Trigonometry Examples

$\frac{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}{\csc (x)} = \sin (x)$

Step 1

Start on the left side.

$\frac{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}{\csc (x)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.2

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

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

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4

Expand $(\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) (\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x)})$ using the FOIL Method.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Apply the distributive property.

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

Step 2.4.3

Apply the distributive property.

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

Step 2.5

Combine the opposite terms in $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)}) + \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})$ .

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Step 2.5.1

Reorder the factors in the terms $\frac{1}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})$ and $\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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

Step 2.5.2

Add $- \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ and $\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

$(\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)} + 0 + \frac{\sin (x)}{\cos (x)} (- \frac{\sin (x)}{\cos (x)})) \sin (x)$

Step 2.5.3

Add $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}$ and $0$ .

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

Step 2.6

Simplify each term.

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Step 2.6.1

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

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Step 2.6.1.1

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

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

Step 2.6.1.2

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

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

Step 2.6.1.3

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

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

Step 2.6.1.4

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

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

Step 2.6.1.5

Add $1$ and $1$ .

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

Step 2.6.2

Rewrite using the commutative property of multiplication.

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

Step 2.6.3

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

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Step 2.6.3.1

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

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

Step 2.6.3.2

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

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

Step 2.6.3.3

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

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

Step 2.6.3.4

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

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

Step 2.6.3.5

Add $1$ and $1$ .

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

Step 2.6.3.6

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

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

Step 2.6.3.7

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

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

Step 2.6.3.8

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

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

Step 2.6.3.9

Add $1$ and $1$ .

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Apply pythagorean identity.

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

Step 2.9

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

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Step 2.9.1

Cancel the common factor.

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

Step 2.9.2

Rewrite the expression.

$1 \sin (x)$

Step 2.10

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

$\sin (x)$

Step 3

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

$\frac{(\sec (x) + \tan (x)) (\sec (x) - \tan (x))}{\csc (x)} = \sin (x)$ is an identity