Trigonometry Examples

$\frac{1 - \cot^{2} (x)}{1 + \cot^{2} (x)} + 2 \cos^{2} (x) = 1$

Step 1

Start on the left side.

$\frac{1 - \cot^{2} (x)}{1 + \cot^{2} (x)} + 2 \cos^{2} (x)$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply pythagorean identity.

$\frac{1 - \cot^{2} (x)}{\csc^{2} (x)} + 2 \cos^{2} (x)$

Step 2.1.2

Simplify the numerator.

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Step 2.1.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2} - \cot^{2} (x)}{\csc^{2} (x)} + 2 \cos^{2} (x)$

Step 2.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cot (x)$ .

$\frac{(1 + \cot (x)) (1 - \cot (x))}{\csc^{2} (x)} + 2 \cos^{2} (x)$

Step 2.1.2.3

Simplify.

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Step 2.1.2.3.1

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{(1 + \frac{\cos (x)}{\sin (x)}) (1 - \cot (x))}{\csc^{2} (x)} + 2 \cos^{2} (x)$

Step 2.1.2.3.2

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{(1 + \frac{\cos (x)}{\sin (x)}) (1 - \frac{\cos (x)}{\sin (x)})}{\csc^{2} (x)} + 2 \cos^{2} (x)$

Step 2.1.3

Simplify the denominator.

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Step 2.1.3.1

Rewrite $\csc (x)$ in terms of sines and cosines.

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

Step 2.1.3.2

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 2.1.3.3

One to any power is one.

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

Step 2.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.5

Expand $(1 + \frac{\cos (x)}{\sin (x)}) (1 - \frac{\cos (x)}{\sin (x)})$ using the FOIL Method.

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Step 2.1.5.1

Apply the distributive property.

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

Step 2.1.5.2

Apply the distributive property.

$(1 \cdot 1 + 1 (- \frac{\cos (x)}{\sin (x)}) + \frac{\cos (x)}{\sin (x)} (1 - \frac{\cos (x)}{\sin (x)})) \sin^{2} (x) + 2 \cos^{2} (x)$

Step 2.1.5.3

Apply the distributive property.

$(1 \cdot 1 + 1 (- \frac{\cos (x)}{\sin (x)}) + \frac{\cos (x)}{\sin (x)} \cdot 1 + \frac{\cos (x)}{\sin (x)} (- \frac{\cos (x)}{\sin (x)})) \sin^{2} (x) + 2 \cos^{2} (x)$

Step 2.1.6

Simplify and combine like terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

Multiply $1$ by $1$ .

$(1 + 1 (- \frac{\cos (x)}{\sin (x)}) + \frac{\cos (x)}{\sin (x)} \cdot 1 + \frac{\cos (x)}{\sin (x)} (- \frac{\cos (x)}{\sin (x)})) \sin^{2} (x) + 2 \cos^{2} (x)$

Step 2.1.6.1.2

Multiply $- \frac{\cos (x)}{\sin (x)}$ by $1$ .

$(1 - \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)} \cdot 1 + \frac{\cos (x)}{\sin (x)} (- \frac{\cos (x)}{\sin (x)})) \sin^{2} (x) + 2 \cos^{2} (x)$

Step 2.1.6.1.3

Multiply $\frac{\cos (x)}{\sin (x)}$ by $1$ .

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

Step 2.1.6.1.4

Rewrite using the commutative property of multiplication.

$(1 - \frac{\cos (x)}{\sin (x)} + \frac{\cos (x)}{\sin (x)} - \frac{\cos (x)}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}) \sin^{2} (x) + 2 \cos^{2} (x)$

Step 2.1.6.1.5

Multiply $- \frac{\cos (x)}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)}$ .

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Step 2.1.6.1.5.1

Multiply $\frac{\cos (x)}{\sin (x)}$ by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 2.1.6.1.5.2

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

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

Step 2.1.6.1.5.3

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

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

Step 2.1.6.1.5.4

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

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

Step 2.1.6.1.5.5

Add $1$ and $1$ .

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

Step 2.1.6.1.5.6

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.1.6.1.5.7

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.1.6.1.5.8

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

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

Step 2.1.6.1.5.9

Add $1$ and $1$ .

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

Step 2.1.6.2

Add $- \frac{\cos (x)}{\sin (x)}$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 2.1.6.3

Add $1$ and $0$ .

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

Step 2.1.7

Apply the distributive property.

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

Step 2.1.8

Multiply $\sin^{2} (x)$ by $1$ .

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

Step 2.1.9

Cancel the common factor of $\sin^{2} (x)$ .

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Step 2.1.9.1

Move the leading negative in $- \frac{\cos^{2} (x)}{\sin^{2} (x)}$ into the numerator.

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

Step 2.1.9.2

Cancel the common factor.

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

Step 2.1.9.3

Rewrite the expression.

$\sin^{2} (x) - \cos^{2} (x) + 2 \cos^{2} (x)$

Step 2.2

Add $- \cos^{2} (x)$ and $2 \cos^{2} (x)$ .

$\sin^{2} (x) + \cos^{2} (x)$

Step 2.3

Apply pythagorean identity.

$1$

Step 3

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{1 - \cot^{2} (x)}{1 + \cot^{2} (x)} + 2 \cos^{2} (x) = 1$ is an identity