Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Simplify.

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Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

Subtract $1$ from $1$ .

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of ${\cos (x)}^{2}$ and $\cos (x)$ .

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Step 3.4.1

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

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

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Multiply by $1$ .

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

Step 3.4.2.2

Cancel the common factor.

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

Step 3.4.2.3

Rewrite the expression.

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

Step 3.4.2.4

Divide $- \cos (x)$ by $1$ .

$\cos (x) + 1 - \cos (x)$

Step 3.5

Subtract $\cos (x)$ from $\cos (x)$ .

$0 + 1$

Step 3.6

Add $0$ and $1$ .

$1$

Step 4

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

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

Step 5

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

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