Trigonometry Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

Simplify the denominator.

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Step 2.1

Rearrange terms.

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

Step 2.2

Apply pythagorean identity.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

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

Step 7

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

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

Step 8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.1

Apply the product rule to $- \sin (x) \cos (x)$ .

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

Step 8.2

Apply the product rule to $- \sin (x)$ .

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

Step 9

Raise $- 1$ to the power of $2$ .

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

Step 10

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

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