Trigonometry Examples

$\cot^{2} (x) + \csc^{2} (x) = 1 + 2 \cot^{2} (x)$

Step 1

Start on the left side.

$\cot^{2} (x) + \csc^{2} (x)$

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Add ${\csc (x)}^{2}$ and ${\csc (x)}^{2}$ .

$2 \csc^{2} (x) - 1$

Step 4

Apply Pythagorean identity in reverse.

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

Step 5

Convert to sines and cosines.

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Multiply $2$ by $1$ .

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

Step 6.1.3

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

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

Step 6.2

Subtract $1$ from $2$ .

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

Step 7

Rewrite $1 + \frac{2 \cos^{2} (x)}{\sin^{2} (x)}$ as $1 + 2 \cot^{2} (x)$ .

$1 + 2 \cot^{2} (x)$

Step 8

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

$\cot^{2} (x) + \csc^{2} (x) = 1 + 2 \cot^{2} (x)$ is an identity