Trigonometry Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

Simplify with factoring out.

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Step 2.1

Reorder $1$ and $- \sec^{2} (x)$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Apply pythagorean identity.

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

Step 4

Rewrite using the commutative property of multiplication.

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

Step 5

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

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

Step 6

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

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

Step 7

Cancel the common factor of $\cos^{2} (x)$ .

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Step 7.1

Factor $\cos^{2} (x)$ out of $- \cos^{2} (x)$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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