Trigonometry Examples

$1 - \sec (x) \cos^{3} (x) = \sin^{2} (x)$

Step 1

Start on the left side.

$1 - \sec (x) \cos^{3} (x)$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$1 - \frac{1}{\cos (x)} \cos^{3} (x)$

Step 2.1.2

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

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Step 2.1.2.1

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

$1 + \frac{- 1}{\cos (x)} \cos^{3} (x)$

Step 2.1.2.2

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

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

Step 2.1.2.3

Cancel the common factor.

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

Step 2.1.2.4

Rewrite the expression.

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

Step 2.1.3

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

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

Step 2.2

Apply pythagorean identity.

$\sin^{2} (x)$

Step 3

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

$1 - \sec (x) \cos^{3} (x) = \sin^{2} (x)$ is an identity