Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Apply pythagorean identity.

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

One to any power is one.

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

Step 4.1.2

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

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

Step 4.1.3

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

Tap for more steps...

Step 4.1.3.1

Cancel the common factor.

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

Step 4.1.3.2

Rewrite the expression.

$2 - 1 \cdot 1$

Step 4.1.4

Multiply $- 1$ by $1$ .

$2 - 1$

Step 4.2

Subtract $1$ from $2$ .

$1$

Step 5

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

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