Simplify (1-sec(theta)^2)/(sec(theta)^2)

$\frac{1 - \sec^{2} (θ)}{\sec^{2} (θ)}$

Step 1

Simplify with factoring out.

Reorder $1$ and $- \sec^{2} (θ)$ .

$\frac{- \sec^{2} (θ) + 1}{\sec^{2} (θ)}$

Factor $- 1$ out of $- \sec^{2} (θ)$ .

$\frac{- (\sec^{2} (θ)) + 1}{\sec^{2} (θ)}$

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- \sec^{2} (θ) - 1 \cdot - 1}{\sec^{2} (θ)}$

Factor $- 1$ out of $- \sec^{2} (θ) - 1 \cdot - 1$ .

$\frac{- (\sec^{2} (θ) - 1)}{\sec^{2} (θ)}$

Step 2

Apply pythagorean identity.

$\frac{- \tan^{2} (θ)}{\sec^{2} (θ)}$

Step 3

Simplify with factoring out.

Move the negative in front of the fraction.

$- \frac{\tan^{2} (θ)}{\sec^{2} (θ)}$

Rewrite $\frac{\tan^{2} (θ)}{\sec^{2} (θ)}$ as ${(\frac{\tan (θ)}{\sec (θ)})}^{2}$ .

$- {(\frac{\tan (θ)}{\sec (θ)})}^{2}$

Step 4

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

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

Step 5

Rewrite $\tan (θ)$ in terms of sines and cosines.

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

Step 6

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

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

Step 7

Write $\cos (θ)$ as a fraction with denominator $1$ .

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

Step 8

Cancel the common factor of $\cos (θ)$ .

Cancel the common factor.

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

Rewrite the expression.

$- \sin^{2} (θ)$