Trigonometry Examples

$\frac{2 + \csc (A)}{\sec (A)} - 2 \cos (A) = \cot (A)$

Step 1

Start on the left side.

$\frac{2 + \csc (A)}{\sec (A)} - 2 \cos (A)$

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to $\csc (A)$ .

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

Step 2.2

Apply the reciprocal identity to $\sec (A)$ .

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

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Combine $\frac{1}{\sin (A)}$ and $\cos (A)$ .

$2 \cos (A) + \frac{\cos (A)}{\sin (A)} - 2 \cos (A)$

Step 3.2

Subtract $2 \cos (A)$ from $2 \cos (A)$ .

$0 + \frac{\cos (A)}{\sin (A)}$

Step 3.3

Add $0$ and $\frac{\cos (A)}{\sin (A)}$ .

$\frac{\cos (A)}{\sin (A)}$

Step 4

Rewrite $\frac{\cos (A)}{\sin (A)}$ as $\cot (A)$ .

$\cot (A)$

Step 5

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

$\frac{2 + \csc (A)}{\sec (A)} - 2 \cos (A) = \cot (A)$ is an identity