Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Simplify the expression.

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Step 2.1

Rearrange terms.

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

Step 2.2

Apply pythagorean identity.

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

Step 2.3

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

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

Step 2.4

Rewrite $\frac{1^{2}}{\sec^{2} (x)}$ as ${(\frac{1}{\sec (x)})}^{2}$ .

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

Step 2.5

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

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

Step 2.6

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

${(1 \cos (x))}^{2}$

Step 2.7

Multiply $\cos (x)$ by $1$ .

$\cos^{2} (x)$

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Rewrite $1 - \sin^{2} (x)$ as $(1 - \sin (x)) (1 + \sin (x))$ .

$(1 - \sin (x)) (1 + \sin (x))$

Step 5

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

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