Trigonometry Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

Apply pythagorean identity.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

$\frac{\cos (x) (- \cos (x))}{\cos (x)}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.2.4

Divide $- \cos (x)$ by $1$ .

$- \cos (x)$