Trigonometry Examples

$\tan (\frac{a}{2}) = - \frac{\sqrt{1 - \cos (a)}}{1 + \cos (a)}$

Step 1

Move all terms containing $a$ to the left side of the equation.

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Step 1.1

Add $\frac{\sqrt{1 - \cos (a)}}{1 + \cos (a)}$ to both sides of the equation.

$\tan (\frac{a}{2}) + \frac{\sqrt{1 - \cos (a)}}{1 + \cos (a)} = 0$

Step 1.2

Rewrite $\tan (\frac{a}{2})$ in terms of sines and cosines.

$\frac{\sin (\frac{a}{2})}{\cos (\frac{a}{2})} + \frac{\sqrt{1 - \cos (a)}}{1 + \cos (a)} = 0$

Step 1.3

To write $\frac{\sin (\frac{a}{2})}{\cos (\frac{a}{2})}$ as a fraction with a common denominator, multiply by $\frac{1 + \cos (a)}{1 + \cos (a)}$ .

$\frac{\sin (\frac{a}{2})}{\cos (\frac{a}{2})} \cdot \frac{1 + \cos (a)}{1 + \cos (a)} + \frac{\sqrt{1 - \cos (a)}}{1 + \cos (a)} = 0$

Step 1.4

To write $\frac{\sqrt{1 - \cos (a)}}{1 + \cos (a)}$ as a fraction with a common denominator, multiply by $\frac{\cos (\frac{a}{2})}{\cos (\frac{a}{2})}$ .

$\frac{\sin (\frac{a}{2})}{\cos (\frac{a}{2})} \cdot \frac{1 + \cos (a)}{1 + \cos (a)} + \frac{\sqrt{1 - \cos (a)}}{1 + \cos (a)} \cdot \frac{\cos (\frac{a}{2})}{\cos (\frac{a}{2})} = 0$

Step 1.5

Write each expression with a common denominator of $\cos (\frac{a}{2}) (1 + \cos (a))$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{\sin (\frac{a}{2})}{\cos (\frac{a}{2})}$ by $\frac{1 + \cos (a)}{1 + \cos (a)}$ .

$\frac{\sin (\frac{a}{2}) (1 + \cos (a))}{\cos (\frac{a}{2}) (1 + \cos (a))} + \frac{\sqrt{1 - \cos (a)}}{1 + \cos (a)} \cdot \frac{\cos (\frac{a}{2})}{\cos (\frac{a}{2})} = 0$

Step 1.5.2

Multiply $\frac{\sqrt{1 - \cos (a)}}{1 + \cos (a)}$ by $\frac{\cos (\frac{a}{2})}{\cos (\frac{a}{2})}$ .

$\frac{\sin (\frac{a}{2}) (1 + \cos (a))}{\cos (\frac{a}{2}) (1 + \cos (a))} + \frac{\sqrt{1 - \cos (a)} \cos (\frac{a}{2})}{(1 + \cos (a)) \cos (\frac{a}{2})} = 0$

Step 1.5.3

Reorder the factors of $(1 + \cos (a)) \cos (\frac{a}{2})$ .

$\frac{\sin (\frac{a}{2}) (1 + \cos (a))}{\cos (\frac{a}{2}) (1 + \cos (a))} + \frac{\sqrt{1 - \cos (a)} \cos (\frac{a}{2})}{\cos (\frac{a}{2}) (1 + \cos (a))} = 0$

Step 1.6

Combine the numerators over the common denominator.

$\frac{\sin (\frac{a}{2}) (1 + \cos (a)) + \sqrt{1 - \cos (a)} \cos (\frac{a}{2})}{\cos (\frac{a}{2}) (1 + \cos (a))} = 0$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Apply the distributive property.

$\frac{\sin (\frac{a}{2}) \cdot 1 + \sin (\frac{a}{2}) \cos (a) + \sqrt{1 - \cos (a)} \cos (\frac{a}{2})}{\cos (\frac{a}{2}) (1 + \cos (a))} = 0$

Step 1.7.2

Multiply $\sin (\frac{a}{2})$ by $1$ .

$\frac{\sin (\frac{a}{2}) + \sin (\frac{a}{2}) \cos (a) + \sqrt{1 - \cos (a)} \cos (\frac{a}{2})}{\cos (\frac{a}{2}) (1 + \cos (a))} = 0$

Step 2

Set the numerator equal to zero.

$\sin (\frac{a}{2}) + \sin (\frac{a}{2}) \cos (a) + \sqrt{1 - \cos (a)} \cos (\frac{a}{2}) = 0$

Step 3

Solve the equation for $a$ .

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Step 3.1

Move all terms not containing $\sqrt{1 - \cos (a)} \cos (\frac{a}{2})$ to the right side of the equation.

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Step 3.1.1

Subtract $\sin (\frac{a}{2})$ from both sides of the equation.

$\sin (\frac{a}{2}) \cos (a) + \sqrt{1 - \cos (a)} \cos (\frac{a}{2}) = - \sin (\frac{a}{2})$

Step 3.1.2

Subtract $\sin (\frac{a}{2}) \cos (a)$ from both sides of the equation.

$\sqrt{1 - \cos (a)} \cos (\frac{a}{2}) = - \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a)$

Step 3.2

To remove the radical on the left side of the equation, square both sides of the equation.

${(\sqrt{1 - \cos (a)} \cos (\frac{a}{2}))}^{2} = {(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - \cos (a)}$ as ${(1 - \cos (a))}^{\frac{1}{2}}$ .

${({(1 - \cos (a))}^{\frac{1}{2}} \cos (\frac{a}{2}))}^{2} = {(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(1 - \cos (a))}^{\frac{1}{2}} \cos (\frac{a}{2}))}^{2}$ .

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Step 3.3.2.1.1

Apply the product rule to ${(1 - \cos (a))}^{\frac{1}{2}} \cos (\frac{a}{2})$ .

${({(1 - \cos (a))}^{\frac{1}{2}})}^{2} \cos^{2} (\frac{a}{2}) = {(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$

Step 3.3.2.1.2

Multiply the exponents in ${({(1 - \cos (a))}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(1 - \cos (a))}^{\frac{1}{2} \cdot 2} \cos^{2} (\frac{a}{2}) = {(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$

Step 3.3.2.1.2.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.2.2.1

Cancel the common factor.

${(1 - \cos (a))}^{\frac{1}{2} \cdot 2} \cos^{2} (\frac{a}{2}) = {(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$

Step 3.3.2.1.2.2.2

Rewrite the expression.

${(1 - \cos (a))}^{1} \cos^{2} (\frac{a}{2}) = {(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$

Step 3.3.2.1.3

Simplify.

$(1 - \cos (a)) \cos^{2} (\frac{a}{2}) = {(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$

Step 3.3.2.1.4

Apply the distributive property.

$1 \cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = {(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$

Step 3.3.2.1.5

Multiply $\cos^{2} (\frac{a}{2})$ by $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = {(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify ${(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$ .

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Step 3.3.3.1.1

Rewrite ${(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))}^{2}$ as $(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a)) (- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = (- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a)) (- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.2

Expand $(- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a)) (- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))$ using the FOIL Method.

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Step 3.3.3.1.2.1

Apply the distributive property.

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = - \sin (\frac{a}{2}) (- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a)) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.2.2

Apply the distributive property.

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = - \sin (\frac{a}{2}) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) (- \sin (\frac{a}{2}) \cos (a)) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.2.3

Apply the distributive property.

Step 3.3.3.1.3

Simplify and combine like terms.

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Step 3.3.3.1.3.1

Simplify each term.

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Step 3.3.3.1.3.1.1

Multiply $- \sin (\frac{a}{2}) (- \sin (\frac{a}{2}))$ .

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Step 3.3.3.1.3.1.1.1

Multiply $- 1$ by $- 1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = 1 \sin (\frac{a}{2}) \sin (\frac{a}{2}) - \sin (\frac{a}{2}) (- \sin (\frac{a}{2}) \cos (a)) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.1.2

Multiply $\sin (\frac{a}{2})$ by $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin (\frac{a}{2}) \sin (\frac{a}{2}) - \sin (\frac{a}{2}) (- \sin (\frac{a}{2}) \cos (a)) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.1.3

Raise $\sin (\frac{a}{2})$ to the power of $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{1} (\frac{a}{2}) \sin (\frac{a}{2}) - \sin (\frac{a}{2}) (- \sin (\frac{a}{2}) \cos (a)) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.1.4

Raise $\sin (\frac{a}{2})$ to the power of $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{1} (\frac{a}{2}) \sin^{1} (\frac{a}{2}) - \sin (\frac{a}{2}) (- \sin (\frac{a}{2}) \cos (a)) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = {\sin (\frac{a}{2})}^{1 + 1} - \sin (\frac{a}{2}) (- \sin (\frac{a}{2}) \cos (a)) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.1.6

Add $1$ and $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) - \sin (\frac{a}{2}) (- \sin (\frac{a}{2}) \cos (a)) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.2

Multiply $- \sin (\frac{a}{2}) (- \sin (\frac{a}{2}) \cos (a))$ .

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Step 3.3.3.1.3.1.2.1

Multiply $- 1$ by $- 1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + 1 \sin (\frac{a}{2}) (\sin (\frac{a}{2}) \cos (a)) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.2.2

Multiply $\sin (\frac{a}{2})$ by $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin (\frac{a}{2}) (\sin (\frac{a}{2}) \cos (a)) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.2.3

Raise $\sin (\frac{a}{2})$ to the power of $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{1} (\frac{a}{2}) \sin (\frac{a}{2}) \cos (a) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.2.4

Raise $\sin (\frac{a}{2})$ to the power of $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{1} (\frac{a}{2}) \sin^{1} (\frac{a}{2}) \cos (a) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + {\sin (\frac{a}{2})}^{1 + 1} \cos (a) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.2.6

Add $1$ and $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2})) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.3

Multiply $- \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}))$ .

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Step 3.3.3.1.3.1.3.1

Multiply $- 1$ by $- 1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + 1 \sin (\frac{a}{2}) \cos (a) \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.3.2

Multiply $\sin (\frac{a}{2})$ by $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin (\frac{a}{2}) \cos (a) \sin (\frac{a}{2}) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.3.3

Raise $\sin (\frac{a}{2})$ to the power of $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{1} (\frac{a}{2}) \sin (\frac{a}{2}) \cos (a) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.3.4

Raise $\sin (\frac{a}{2})$ to the power of $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{1} (\frac{a}{2}) \sin^{1} (\frac{a}{2}) \cos (a) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + {\sin (\frac{a}{2})}^{1 + 1} \cos (a) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.3.6

Add $1$ and $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos (a) - \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.4

Multiply $- \sin (\frac{a}{2}) \cos (a) (- \sin (\frac{a}{2}) \cos (a))$ .

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Step 3.3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos (a) + 1 \sin (\frac{a}{2}) \cos (a) (\sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.4.2

Multiply $\sin (\frac{a}{2})$ by $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin (\frac{a}{2}) \cos (a) (\sin (\frac{a}{2}) \cos (a))$

Step 3.3.3.1.3.1.4.3

Raise $\sin (\frac{a}{2})$ to the power of $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{1} (\frac{a}{2}) \sin (\frac{a}{2}) \cos (a) \cos (a)$

Step 3.3.3.1.3.1.4.4

Raise $\sin (\frac{a}{2})$ to the power of $1$ .

Step 3.3.3.1.3.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos (a) + {\sin (\frac{a}{2})}^{1 + 1} \cos (a) \cos (a)$

Step 3.3.3.1.3.1.4.6

Add $1$ and $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos (a) \cos (a)$

Step 3.3.3.1.3.1.4.7

Raise $\cos (a)$ to the power of $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) (\cos^{1} (a) \cos (a))$

Step 3.3.3.1.3.1.4.8

Raise $\cos (a)$ to the power of $1$ .

Step 3.3.3.1.3.1.4.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) {\cos (a)}^{1 + 1}$

Step 3.3.3.1.3.1.4.10

Add $1$ and $1$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos^{2} (a)$

Step 3.3.3.1.3.2

Add $\sin^{2} (\frac{a}{2}) \cos (a)$ and $\sin^{2} (\frac{a}{2}) \cos (a)$ .

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) = \sin^{2} (\frac{a}{2}) + 2 \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos^{2} (a)$

Step 3.4

Solve for $a$ .

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Step 3.4.1

Move all the expressions to the left side of the equation.

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Step 3.4.1.1

Subtract $\sin^{2} (\frac{a}{2})$ from both sides of the equation.

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) - \sin^{2} (\frac{a}{2}) = 2 \sin^{2} (\frac{a}{2}) \cos (a) + \sin^{2} (\frac{a}{2}) \cos^{2} (a)$

Step 3.4.1.2

Subtract $2 \sin^{2} (\frac{a}{2}) \cos (a)$ from both sides of the equation.

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) - \sin^{2} (\frac{a}{2}) - 2 \sin^{2} (\frac{a}{2}) \cos (a) = \sin^{2} (\frac{a}{2}) \cos^{2} (a)$

Step 3.4.1.3

Subtract $\sin^{2} (\frac{a}{2}) \cos^{2} (a)$ from both sides of the equation.

$\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) - \sin^{2} (\frac{a}{2}) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2

Simplify $\cos^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) - \sin^{2} (\frac{a}{2}) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a)$ .

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Step 3.4.2.1

Move $- \sin^{2} (\frac{a}{2})$ .

$\cos^{2} (\frac{a}{2}) - \sin^{2} (\frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2.2

Apply the cosine double-angle identity.

$\cos (2 \frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2.3

Cancel the common factor of $2$ .

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Step 3.4.2.3.1

Cancel the common factor.

$\cos (2 \frac{a}{2}) - \cos (a) \cos^{2} (\frac{a}{2}) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2.3.2

Rewrite the expression.

$\cos (a) - \cos (a) \cos^{2} (\frac{a}{2}) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2.4

Multiply by $1$ .

$\cos (a) \cdot 1 - \cos (a) \cos^{2} (\frac{a}{2}) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2.5

Factor $\cos (a)$ out of $- \cos (a) \cos^{2} (\frac{a}{2})$ .

$\cos (a) \cdot 1 + \cos (a) (- 1 \cos^{2} (\frac{a}{2})) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2.6

Factor $\cos (a)$ out of $\cos (a) \cdot 1 + \cos (a) (- 1 \cos^{2} (\frac{a}{2}))$ .

$\cos (a) \cdot (1 - 1 \cos^{2} (\frac{a}{2})) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2.7

Rewrite $- 1 \cos^{2} (\frac{a}{2})$ as $- \cos^{2} (\frac{a}{2})$ .

$\cos (a) \cdot (1 - \cos^{2} (\frac{a}{2})) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2.8

Apply pythagorean identity.

$\cos (a) \cdot \sin^{2} (\frac{a}{2}) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2.9

Reorder the factors of $\cos (a) \sin^{2} (\frac{a}{2})$ .

$\sin^{2} (\frac{a}{2}) \cos (a) - 2 \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.2.10

Subtract $2 \sin^{2} (\frac{a}{2}) \cos (a)$ from $\sin^{2} (\frac{a}{2}) \cos (a)$ .

$- \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.3

Factor $- \cos (a) \sin^{2} (\frac{a}{2})$ out of $- \sin^{2} (\frac{a}{2}) \cos (a) - \sin^{2} (\frac{a}{2}) \cos^{2} (a)$ .

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Step 3.4.3.1

Reorder the expression.

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Step 3.4.3.1.1

Move $\sin^{2} (\frac{a}{2})$ .

$- 1 \cos (a) \sin^{2} (\frac{a}{2}) - \sin^{2} (\frac{a}{2}) \cos^{2} (a) = 0$

Step 3.4.3.1.2

Move $\sin^{2} (\frac{a}{2})$ .

$- 1 \cos (a) \sin^{2} (\frac{a}{2}) - 1 \cos^{2} (a) \sin^{2} (\frac{a}{2}) = 0$

Step 3.4.3.1.3

Reorder $- 1 \cos (a) \sin^{2} (\frac{a}{2})$ and $- 1 \cos^{2} (a) \sin^{2} (\frac{a}{2})$ .

$- 1 \cos^{2} (a) \sin^{2} (\frac{a}{2}) - 1 \cos (a) \sin^{2} (\frac{a}{2}) = 0$

Step 3.4.3.2

Factor $- \cos (a) \sin^{2} (\frac{a}{2})$ out of $- 1 \cos^{2} (a) \sin^{2} (\frac{a}{2})$ .

$- \cos (a) \sin^{2} (\frac{a}{2}) \cos (a) - 1 \cos (a) \sin^{2} (\frac{a}{2}) = 0$

Step 3.4.3.3

Factor $- \cos (a) \sin^{2} (\frac{a}{2})$ out of $- 1 \cos (a) \sin^{2} (\frac{a}{2})$ .

$- \cos (a) \sin^{2} (\frac{a}{2}) \cos (a) - \cos (a) \sin^{2} (\frac{a}{2}) \cdot 1 = 0$

Step 3.4.3.4

Factor $- \cos (a) \sin^{2} (\frac{a}{2})$ out of $- \cos (a) \sin^{2} (\frac{a}{2}) \cos (a) - \cos (a) \sin^{2} (\frac{a}{2}) \cdot 1$ .

$- \cos (a) \sin^{2} (\frac{a}{2}) (\cos (a) + 1) = 0$

Step 3.4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos (a) = 0$

$\sin^{2} (\frac{a}{2}) = 0$

$\cos (a) + 1 = 0$

Step 3.4.5

Set $\cos (a)$ equal to $0$ and solve for $a$ .

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Step 3.4.5.1

Set $\cos (a)$ equal to $0$ .

$\cos (a) = 0$

Step 3.4.5.2

Solve $\cos (a) = 0$ for $a$ .

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Step 3.4.5.2.1

Take the inverse cosine of both sides of the equation to extract $a$ from inside the cosine.

$a = \arccos (0)$

Step 3.4.5.2.2

Simplify the right side.

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Step 3.4.5.2.2.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$a = \frac{π}{2}$

Step 3.4.5.2.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$a = 2 π - \frac{π}{2}$

Step 3.4.5.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 3.4.5.2.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 3.4.5.2.4.2

Combine fractions.

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Step 3.4.5.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$a = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 3.4.5.2.4.2.2

Combine the numerators over the common denominator.

$a = \frac{2 π \cdot 2 - π}{2}$

Step 3.4.5.2.4.3

Simplify the numerator.

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Step 3.4.5.2.4.3.1

Multiply $2$ by $2$ .

$a = \frac{4 π - π}{2}$

Step 3.4.5.2.4.3.2

Subtract $π$ from $4 π$ .

$a = \frac{3 π}{2}$

Step 3.4.5.2.5

Find the period of $\cos (a)$ .

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Step 3.4.5.2.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.4.5.2.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.4.5.2.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.4.5.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.4.5.2.6

The period of the $\cos (a)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$a = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 3.4.6

Set $\sin^{2} (\frac{a}{2})$ equal to $0$ and solve for $a$ .

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Step 3.4.6.1

Set $\sin^{2} (\frac{a}{2})$ equal to $0$ .

$\sin^{2} (\frac{a}{2}) = 0$

Step 3.4.6.2

Solve $\sin^{2} (\frac{a}{2}) = 0$ for $a$ .

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Step 3.4.6.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sin (\frac{a}{2}) = \pm \sqrt{0}$

Step 3.4.6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 3.4.6.2.2.1

Rewrite $0$ as $0^{2}$ .

$\sin (\frac{a}{2}) = \pm \sqrt{0^{2}}$

Step 3.4.6.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\sin (\frac{a}{2}) = \pm 0$

Step 3.4.6.2.2.3

Plus or minus $0$ is $0$ .

$\sin (\frac{a}{2}) = 0$

Step 3.4.6.2.3

Take the inverse sine of both sides of the equation to extract $a$ from inside the sine.

$\frac{a}{2} = \arcsin (0)$

Step 3.4.6.2.4

Simplify the right side.

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Step 3.4.6.2.4.1

The exact value of $\arcsin (0)$ is $0$ .

$\frac{a}{2} = 0$

Step 3.4.6.2.5

Set the numerator equal to zero.

$a = 0$

Step 3.4.6.2.6

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$\frac{a}{2} = π - 0$

Step 3.4.6.2.7

Solve for $a$ .

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Step 3.4.6.2.7.1

Multiply both sides of the equation by $2$ .

$2 \frac{a}{2} = 2 (π - 0)$

Step 3.4.6.2.7.2

Simplify both sides of the equation.

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Step 3.4.6.2.7.2.1

Simplify the left side.

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Step 3.4.6.2.7.2.1.1

Cancel the common factor of $2$ .

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Step 3.4.6.2.7.2.1.1.1

Cancel the common factor.

$2 \frac{a}{2} = 2 (π - 0)$

Step 3.4.6.2.7.2.1.1.2

Rewrite the expression.

$a = 2 (π - 0)$

Step 3.4.6.2.7.2.2

Simplify the right side.

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Step 3.4.6.2.7.2.2.1

Subtract $0$ from $π$ .

$a = 2 π$

Step 3.4.6.2.8

Find the period of $\sin (\frac{a}{2})$ .

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Step 3.4.6.2.8.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.4.6.2.8.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 3.4.6.2.8.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 3.4.6.2.8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 3.4.6.2.8.5

Multiply $2$ by $2$ .

$4 π$

Step 3.4.6.2.9

The period of the $\sin (\frac{a}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$a = 4 π n, 2 π + 4 π n$ , for any integer $n$

Step 3.4.7

Set $\cos (a) + 1$ equal to $0$ and solve for $a$ .

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Step 3.4.7.1

Set $\cos (a) + 1$ equal to $0$ .

$\cos (a) + 1 = 0$

Step 3.4.7.2

Solve $\cos (a) + 1 = 0$ for $a$ .

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Step 3.4.7.2.1

Subtract $1$ from both sides of the equation.

$\cos (a) = - 1$

Step 3.4.7.2.2

Take the inverse cosine of both sides of the equation to extract $a$ from inside the cosine.

$a = \arccos (- 1)$

Step 3.4.7.2.3

Simplify the right side.

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Step 3.4.7.2.3.1

The exact value of $\arccos (- 1)$ is $π$ .

$a = π$

Step 3.4.7.2.4

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$a = 2 π - π$

Step 3.4.7.2.5

Subtract $π$ from $2 π$ .

$a = π$

Step 3.4.7.2.6

Find the period of $\cos (a)$ .

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Step 3.4.7.2.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.4.7.2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.4.7.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.4.7.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.4.7.2.7

The period of the $\cos (a)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$a = π + 2 π n$ , for any integer $n$

Step 3.4.8

The final solution is all the values that make $- \cos (a) \sin^{2} (\frac{a}{2}) (\cos (a) + 1) = 0$ true.

$a = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n, 4 π n, 2 π + 4 π n, π + 2 π n$ , for any integer $n$

Step 4

Consolidate the answers.

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Step 4.1

Consolidate $\frac{π}{2} + 2 π n$ and $\frac{3 π}{2} + 2 π n$ to $\frac{π}{2} + π n$ .

$a = \frac{π}{2} + π n, 4 π n, 2 π + 4 π n, π + 2 π n$ , for any integer $n$

Step 4.2

Consolidate $4 π n$ and $2 π + 4 π n$ to $2 π n$ .

$a = \frac{π}{2} + π n, 2 π n, π + 2 π n$ , for any integer $n$

Step 4.3

Consolidate $2 π n$ and $π + 2 π n$ to $π n$ .

$a = \frac{π}{2} + π n, π n$ , for any integer $n$

Step 4.4

Consolidate the answers.

$a = \frac{π n}{2}$ , for any integer $n$

Step 5

Verify each of the solutions by substituting them into $\tan (\frac{a}{2}) = - \frac{\sqrt{1 - \cos (a)}}{1 + \cos (a)}$ and solving.

$a = \frac{3 π}{2} + 2 π n, 2 π n$ , for any integer $n$