Trigonometry Examples

$\frac{\sec (B)}{\cos (B)} - \frac{\tan (B)}{\cot (B)} = 1$

Step 1

Start on the left side.

$\frac{\sec (B)}{\cos (B)} - \frac{\tan (B)}{\cot (B)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\frac{\frac{1}{\cos (B)}}{\cos (B)} - \frac{\tan (B)}{\cot (B)}$

Step 2.1.2

Rewrite $\frac{\frac{1}{\cos (B)}}{\cos (B)}$ as a product.

$\frac{1}{\cos (B)} \cdot \frac{1}{\cos (B)} - \frac{\tan (B)}{\cot (B)}$

Step 2.1.3

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

$\frac{1}{\cos (B) \cos (B)} - \frac{\tan (B)}{\cot (B)}$

Step 2.1.4

Simplify the denominator.

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Step 2.1.4.1

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

$\frac{1}{\cos^{1} (B) \cos (B)} - \frac{\tan (B)}{\cot (B)}$

Step 2.1.4.2

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

$\frac{1}{\cos^{1} (B) \cos^{1} (B)} - \frac{\tan (B)}{\cot (B)}$

Step 2.1.4.3

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

$\frac{1}{{\cos (B)}^{1 + 1}} - \frac{\tan (B)}{\cot (B)}$

Step 2.1.4.4

Add $1$ and $1$ .

$\frac{1}{\cos^{2} (B)} - \frac{\tan (B)}{\cot (B)}$

Step 2.1.5

Rewrite $\cot (B)$ in terms of sines and cosines.

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

Step 2.1.6

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

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

Step 2.1.7

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (B)}{\sin (B)}$ .

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

Step 2.1.8

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

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

Step 2.1.9

Simplify the numerator.

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Step 2.1.9.1

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

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

Step 2.1.9.2

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

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

Step 2.1.9.3

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

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

Step 2.1.9.4

Add $1$ and $1$ .

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

Step 2.1.10

Simplify the denominator.

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Step 2.1.10.1

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

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

Step 2.1.10.2

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

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

Step 2.1.10.3

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

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

Step 2.1.10.4

Add $1$ and $1$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Apply pythagorean identity.

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

Step 2.4

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

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

$1$

Step 3

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

$\frac{\sec (B)}{\cos (B)} - \frac{\tan (B)}{\cot (B)} = 1$ is an identity