Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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Step 3.2.1

Factor $\cos (x)$ out of ${\cos (x)}^{2}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

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

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Step 3.3.1

Move the leading negative in $- \frac{1}{\cos (x)}$ into the numerator.

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

Step 3.3.2

Factor $\cos (x)$ out of ${\cos (x)}^{2}$ .

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

Simplify each term.

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Step 3.4.1

Move $- 1$ to the left of $\cos (x)$ .

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

Step 3.4.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 3.5

Expand $(\cos (x) \sin (x) - \cos (x)) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$ using the FOIL Method.

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

Apply the distributive property.

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

Step 3.5.3

Apply the distributive property.

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

Step 3.6

Simplify and combine like terms.

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

Step 4

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

$\cos^{2} (x) (\tan (x) - \sec (x)) (\tan (x) + \sec (x)) = \sin^{2} (x) - 1$ is an identity