Trigonometry Examples

$\csc (A) \sin (2 A) - \sec (A) = \cos (2 A) \sec (A)$

Step 1

Start on the left side.

$\csc (A) \sin (2 A) - \sec (A)$

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

Combine $\frac{1}{\sin (A)}$ and $\sin (2 A)$ .

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

Step 2.3

Apply the sine double-angle identity.

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

Step 2.4

Cancel the common factor of $\sin (A)$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Divide $2 \cos (A)$ by $1$ .

$2 \cos (A) - \sec (A)$

Step 2.5

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

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

Step 3

Write $2 \cos (A)$ as a fraction with denominator $1$ .

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

Step 4

Subtract fractions.

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Step 4.1

To write $\frac{2 \cos (A)}{1}$ as a fraction with a common denominator, multiply by $\frac{\cos (A)}{\cos (A)}$ .

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

Step 4.2

Multiply $\frac{2 \cos (A)}{1}$ by $\frac{\cos (A)}{\cos (A)}$ .

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 5

Multiply $2 \cos (A) \cos (A)$ .

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

Step 6

Apply the cosine double-angle identity.

$\frac{\cos (2 A)}{\cos (A)}$

Step 7

Rewrite $\frac{\cos (2 A)}{\cos (A)}$ as $\cos (2 A) \sec (A)$ .

$\cos (2 A) \sec (A)$

Step 8

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

$\csc (A) \sin (2 A) - \sec (A) = \cos (2 A) \sec (A)$ is an identity