Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

Multiply $\frac{1 - \sin (t)}{\cos^{2} (t)}$ by $\frac{1 - \sin (t)}{1 - \sin (t)}$ .

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

Step 2.3.2

Multiply $\frac{1}{1 - \sin (t)}$ by $\frac{\cos^{2} (t)}{\cos^{2} (t)}$ .

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

Step 2.3.3

Reorder the factors of $(1 - \sin (t)) \cos^{2} (t)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Expand $(1 - \sin (t)) (1 - \sin (t))$ using the FOIL Method.

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Step 2.5.1.1

Apply the distributive property.

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

Step 2.5.1.2

Apply the distributive property.

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

Step 2.5.1.3

Apply the distributive property.

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

Step 2.5.2

Simplify and combine like terms.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Multiply $1$ by $1$ .

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

Step 2.5.2.1.2

Multiply $- \sin (t)$ by $1$ .

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

Step 2.5.2.1.3

Multiply $- 1$ by $1$ .

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

Step 2.5.2.1.4

Multiply $- \sin (t) (- \sin (t))$ .

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Step 2.5.2.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.2.1.4.2

Multiply $\sin (t)$ by $1$ .

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

Step 2.5.2.1.4.3

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

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

Step 2.5.2.1.4.4

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

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

Step 2.5.2.1.4.5

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

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

Step 2.5.2.1.4.6

Add $1$ and $1$ .

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

Step 2.5.2.2

Subtract $\sin (t)$ from $- \sin (t)$ .

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

Step 2.5.3

Rewrite $1 - 2 \sin (t) + \sin^{2} (t) + \cos^{2} (t)$ in a factored form.

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Step 2.5.3.1

Apply pythagorean identity.

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

Step 2.5.3.2

Add $1$ and $1$ .

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

Step 2.5.3.3

Factor $2$ out of $2 - 2 \sin (t)$ .

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Step 2.5.3.3.1

Factor $2$ out of $2$ .

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

Step 2.5.3.3.2

Factor $2$ out of $- 2 \sin (t)$ .

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

Step 2.5.3.3.3

Factor $2$ out of $2 (1) + 2 (- \sin (t))$ .

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

Step 2.6

Cancel the common factor of $1 - \sin (t)$ .

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Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

$\frac{2}{\cos^{2} (t)}$

Step 3

Rewrite $\frac{2}{\cos^{2} (t)}$ as $2 \sec^{2} (t)$ .

$2 \sec^{2} (t)$

Step 4

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

$\frac{1 - \sin (t)}{\cos^{2} (t)} + \frac{1}{1 - \sin (t)} = 2 \sec^{2} (t)$ is an identity