Trigonometry Examples

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

Step 1

Start on the left side.

$\sin (x) \tan (x) + \cos (x) - \sec (x) + 1$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

Multiply $\sin (x) \frac{\sin (x)}{\cos (x)}$ .

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Step 2.1.2.1

Combine $\sin (x)$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 2.1.2.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.1.2.3

Raise $\sin (x)$ to the power of $1$ .

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

Step 2.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.2.5

Add $1$ and $1$ .

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

Step 2.1.3

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

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

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

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

Step 2.4

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.5

Factor $- 1$ out of $\sin^{2} (x)$ .

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

Step 2.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

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

Step 2.7

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

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

Step 2.8

Apply pythagorean identity.

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

Step 2.9

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

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Step 2.9.1

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

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

Step 2.9.2

Cancel the common factors.

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Step 2.9.2.1

Multiply by $1$ .

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

Step 2.9.2.2

Cancel the common factor.

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

Step 2.9.2.3

Rewrite the expression.

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

Step 2.9.2.4

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

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

Step 2.10

Combine the opposite terms in $\cos (x) + 1 - \cos (x)$ .

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Step 2.10.1

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

$0 + 1$

Step 2.10.2

Add $0$ and $1$ .

$1$

Step 3

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

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

Step 4

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

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