Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity.

Tap for more steps...

Step 2.1

Apply pythagorean identity.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

Tap for more steps...

Step 3.1

Apply the reciprocal identity to $\sec (x)$ .

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

Step 3.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

One to any power is one.

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

Step 4.2

Combine $\frac{1}{{\cos (x)}^{2}}$ and ${\sin (x)}^{2}$ .

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

Step 5

Write $- 1$ as a fraction with denominator $1$ .

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

Step 6

Combine.

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

Step 7

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

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

Step 8

Move the negative in front of the fraction.

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

Step 9

Now consider the right side of the equation.

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

Step 10

Convert to sines and cosines.

Tap for more steps...

Step 10.1

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 10.2

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

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

Step 11

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

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