Trigonometry Examples

$\frac{1}{1 - \sin (x)} = \sec^{2} (x) + \tan (x) \sec (x)$

Step 1

Start on the right side.

$\sec^{2} (x) + \tan (x) \sec (x)$

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to $\sec (x)$ .

${(\frac{1}{\cos (x)})}^{2} + \tan (x) \sec (x)$

Step 2.2

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 2.3

Apply the reciprocal identity to $\sec (x)$ .

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

One to any power is one.

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Apply Pythagorean identity in reverse.

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

Step 6

Simplify.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

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

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

Step 6.1.2

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 $b = \sin (x)$ .

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

Step 6.2

Cancel the common factor of $1 + \sin (x)$ .

$\frac{1}{1 - \sin (x)}$

Step 7

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

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