Trigonometry Examples

$\tan (x) \cot (x) - \cos^{2} (x) = \sin^{2} (x)$

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

Cancel the common factor.

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

Step 4.1.1.2

Rewrite the expression.

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

Step 4.1.2

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

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Step 4.1.2.1

Cancel the common factor.

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

Step 4.1.2.2

Rewrite the expression.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

Multiply $- 1$ by $1$ .

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

Step 4.1.5

Multiply $- - {\sin (x)}^{2}$ .

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Step 4.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.5.2

Multiply ${\sin (x)}^{2}$ by $1$ .

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

Step 4.2

Subtract $1$ from $1$ .

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

Step 4.3

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

$\sin^{2} (x)$

Step 5

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

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