Trigonometry Examples

$\sec (- x) - \sin (- x) \tan (- x) = \cos (x)$

Step 1

Start on the left side.

$\sec (- x) - \sin (- x) \tan (- x)$

Step 2

Since $\sec (- x)$ is an even function, rewrite $\sec (- x)$ as $\sec (x)$ .

$\sec (x) - \sin (- x) \tan (- x)$

Step 3

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$\sec (x) - - \sin (x) \tan (- x)$

Step 4

Since $\tan (- x)$ is an odd function, rewrite $\tan (- x)$ as $- \tan (x)$ .

$\sec (x) - - \sin (x) (- \tan (x))$

Step 5

Simplify the expression.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

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

Step 5.1.2

Multiply $- - \sin (x)$ .

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Step 5.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.2.2

Multiply $\sin (x)$ by $1$ .

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

Step 5.1.3

Rewrite using the commutative property of multiplication.

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

Step 5.1.4

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

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

Step 5.1.5

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

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Step 5.1.5.1

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

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

Step 5.1.5.2

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

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

Step 5.1.5.3

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

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

Step 5.1.5.4

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

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

Step 5.1.5.5

Add $1$ and $1$ .

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 5.3

Apply pythagorean identity.

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

Step 5.4

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

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Step 5.4.1

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

$\frac{\cos (x) \cos (x)}{\cos (x)}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Multiply by $1$ .

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

Step 5.4.2.2

Cancel the common factor.

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

Step 5.4.2.3

Rewrite the expression.

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

Step 5.4.2.4

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

$\cos (x)$

Step 6

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

$\sec (- x) - \sin (- x) \tan (- x) = \cos (x)$ is an identity