Trigonometry Examples

$\tan (x) (\tan (x) + \cot (x)) = \sec^{2} (x)$

Step 1

Start on the left side.

$\tan (x) (\tan (x) + \cot (x))$

Step 2

Apply the distributive property.

$\tan (x) \tan (x) + \tan (x) \cot (x)$

Step 3

Multiply $\tan (x) \tan (x)$ .

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

Step 4

Apply Pythagorean identity in reverse.

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

Step 5

Convert to sines and cosines.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

One to any power is one.

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

Step 6.1.2

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

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Step 6.1.2.1

Cancel the common factor.

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

Step 6.1.2.2

Rewrite the expression.

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

Step 6.1.3

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

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Step 6.1.3.1

Cancel the common factor.

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

Step 6.1.3.2

Rewrite the expression.

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

Step 6.2

Add $- 1$ and $1$ .

$\frac{1}{{\cos (x)}^{2}} + 0$

Step 6.3

Add $\frac{1}{{\cos (x)}^{2}}$ and $0$ .

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

Step 7

Rewrite $\frac{1}{\cos^{2} (x)}$ as $\sec^{2} (x)$ .

$\sec^{2} (x)$

Step 8

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

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