Trigonometry Examples

$\frac{\sec (90 - x)}{\sin (90 - x)} \cdot \frac{\cos (x)}{\tan (90 - x)} - \sec (x)$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\sec (90 - x)$ in terms of sines and cosines.

$\frac{\frac{1}{\cos (90 - x)}}{\sin (90 - x)} \cdot \frac{\cos (x)}{\tan (90 - x)} - \sec (x)$

Step 1.2

Rewrite $\tan (90 - x)$ in terms of sines and cosines.

$\frac{\frac{1}{\cos (90 - x)}}{\sin (90 - x)} \cdot \frac{\cos (x)}{\frac{\sin (90 - x)}{\cos (90 - x)}} - \sec (x)$

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{\cos (90 - x)} \cdot \frac{1}{\sin (90 - x)} \cdot \frac{\cos (x)}{\frac{\sin (90 - x)}{\cos (90 - x)}} - \sec (x)$

Step 1.4

Multiply $\frac{1}{\cos (90 - x)}$ by $\frac{1}{\sin (90 - x)}$ .

$\frac{1}{\cos (90 - x) \sin (90 - x)} \cdot \frac{\cos (x)}{\frac{\sin (90 - x)}{\cos (90 - x)}} - \sec (x)$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{\cos (90 - x) \sin (90 - x)} \cdot (\cos (x) \frac{\cos (90 - x)}{\sin (90 - x)}) - \sec (x)$

Step 1.6

Combine $\cos (x)$ and $\frac{\cos (90 - x)}{\sin (90 - x)}$ .

$\frac{1}{\cos (90 - x) \sin (90 - x)} \cdot \frac{\cos (x) \cos (90 - x)}{\sin (90 - x)} - \sec (x)$

Step 1.7

Cancel the common factor of $\cos (90 - x)$ .

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Step 1.7.1

Factor $\cos (90 - x)$ out of $\cos (90 - x) \sin (90 - x)$ .

$\frac{1}{\cos (90 - x) (\sin (90 - x))} \cdot \frac{\cos (x) \cos (90 - x)}{\sin (90 - x)} - \sec (x)$

Step 1.7.2

Factor $\cos (90 - x)$ out of $\cos (x) \cos (90 - x)$ .

$\frac{1}{\cos (90 - x) (\sin (90 - x))} \cdot \frac{\cos (90 - x) \cos (x)}{\sin (90 - x)} - \sec (x)$

Step 1.7.3

Cancel the common factor.

$\frac{1}{\cos (90 - x) \sin (90 - x)} \cdot \frac{\cos (90 - x) \cos (x)}{\sin (90 - x)} - \sec (x)$

Step 1.7.4

Rewrite the expression.

$\frac{1}{\sin (90 - x)} \cdot \frac{\cos (x)}{\sin (90 - x)} - \sec (x)$

Step 1.8

Multiply $\frac{1}{\sin (90 - x)}$ by $\frac{\cos (x)}{\sin (90 - x)}$ .

$\frac{\cos (x)}{\sin (90 - x) \sin (90 - x)} - \sec (x)$

Step 1.9

Raise $\sin (90 - x)$ to the power of $1$ .

$\frac{\cos (x)}{\sin^{1} (90 - x) \sin (90 - x)} - \sec (x)$

Step 1.10

Raise $\sin (90 - x)$ to the power of $1$ .

$\frac{\cos (x)}{\sin^{1} (90 - x) \sin^{1} (90 - x)} - \sec (x)$

Step 1.11

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

$\frac{\cos (x)}{{\sin (90 - x)}^{1 + 1}} - \sec (x)$

Step 1.12

Add $1$ and $1$ .

$\frac{\cos (x)}{\sin^{2} (90 - x)} - \sec (x)$

Step 1.13

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

$\frac{\cos (x)}{\sin^{2} (90 - x)} - \frac{1}{\cos (x)}$

Step 2

Simplify each term.

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Step 2.1

Factor $\sin (90 - x)$ out of $\sin^{2} (90 - x)$ .

$\frac{\cos (x)}{\sin (90 - x) \sin (90 - x)} - \frac{1}{\cos (x)}$

Step 2.2

Separate fractions.

$\frac{1}{\sin (90 - x)} \cdot \frac{\cos (x)}{\sin (90 - x)} - \frac{1}{\cos (x)}$

Step 2.3

Rewrite $\frac{\cos (x)}{\sin (90 - x)}$ as a product.

$\frac{1}{\sin (90 - x)} (\cos (x) \frac{1}{\sin (90 - x)}) - \frac{1}{\cos (x)}$

Step 2.4

Write $\cos (x)$ as a fraction with denominator $1$ .

$\frac{1}{\sin (90 - x)} (\frac{\cos (x)}{1} \cdot \frac{1}{\sin (90 - x)}) - \frac{1}{\cos (x)}$

Step 2.5

Simplify.

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Step 2.5.1

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

$\frac{1}{\sin (90 - x)} (\cos (x) \frac{1}{\sin (90 - x)}) - \frac{1}{\cos (x)}$

Step 2.5.2

Convert from $\frac{1}{\sin (90 - x)}$ to $\csc (90 - x)$ .

$\frac{1}{\sin (90 - x)} (\cos (x) \csc (90 - x)) - \frac{1}{\cos (x)}$

Step 2.6

Convert from $\frac{1}{\sin (90 - x)}$ to $\csc (90 - x)$ .

$\csc (90 - x) (\cos (x) \csc (90 - x)) - \frac{1}{\cos (x)}$

Step 2.7

Multiply $\csc (90 - x) (\cos (x) \csc (90 - x))$ .

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Step 2.7.1

Raise $\csc (90 - x)$ to the power of $1$ .

$\csc^{1} (90 - x) \csc (90 - x) \cos (x) - \frac{1}{\cos (x)}$

Step 2.7.2

Raise $\csc (90 - x)$ to the power of $1$ .

$\csc^{1} (90 - x) \csc^{1} (90 - x) \cos (x) - \frac{1}{\cos (x)}$

Step 2.7.3

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

${\csc (90 - x)}^{1 + 1} \cos (x) - \frac{1}{\cos (x)}$

Step 2.7.4

Add $1$ and $1$ .

$\csc^{2} (90 - x) \cos (x) - \frac{1}{\cos (x)}$

Step 2.8

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\csc^{2} (90 - x) \cos (x) - \sec (x)$