Trigonometry Examples

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

Step 1

Start on the right side.

$\frac{\cot (x)}{\tan (x)} + 1$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 2.4

Multiply $\frac{\cos (x)}{\sin (x)}$ by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.5.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 2.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.5.4

Add $1$ and $1$ .

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

Step 2.6

Simplify the denominator.

Tap for more steps...

Step 2.6.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.6.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.6.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.6.4

Add $1$ and $1$ .

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (x)$ .

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

Step 4.2

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

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Simplify the numerator.

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

Step 5

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

$\csc^{2} (x)$

Step 6

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

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