Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify.

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Step 4.1

Cancel the common factor of $\cos (x)$ .

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Step 4.1.1

Factor $\cos (x)$ out of $- \cos (x)$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Subtract $1$ from $1$ .

$0 + \frac{{\cos (x)}^{2}}{{\sin (x)}^{2}}$

Step 4.3

Add $0$ and $\frac{{\cos (x)}^{2}}{{\sin (x)}^{2}}$ .

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

Step 5

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

$\cot^{2} (x)$

Step 6

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

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