Trigonometry Examples

$\csc^{2} (x) - \frac{\cot (x)}{\tan (x)}$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 1.4.2

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.4.5

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

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

Step 1.4.6

Add $1$ and $1$ .

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

Step 2

Apply pythagorean identity.

$1$