Trigonometry Examples

$\frac{\cos (\frac{π}{2} - θ)}{\sin (2 θ)} = \frac{\sec (θ)}{2}$

Step 1

Start on the left side.

$\frac{\cos (\frac{π}{2} - θ)}{\sin (2 θ)}$

Step 2

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$\frac{\cos (\frac{π}{2}) \cos (θ) + \sin (\frac{π}{2}) \sin (θ)}{\sin (2 θ)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{0 \cos (θ) + \sin (\frac{π}{2}) \sin (θ)}{\sin (2 θ)}$

Step 3.1.2

Multiply $0$ by $\cos (θ)$ .

$\frac{0 + \sin (\frac{π}{2}) \sin (θ)}{\sin (2 θ)}$

Step 3.1.3

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{0 + 1 \sin (θ)}{\sin (2 θ)}$

Step 3.1.4

Multiply $\sin (θ)$ by $1$ .

$\frac{0 + \sin (θ)}{\sin (2 θ)}$

Step 3.1.5

Add $0$ and $\sin (θ)$ .

$\frac{\sin (θ)}{\sin (2 θ)}$

Step 3.2

Apply the sine double-angle identity.

$\frac{\sin (θ)}{2 \sin (θ) \cos (θ)}$

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

$\frac{\sin (θ)}{2 \sin (θ) \cos (θ)}$

Step 3.3.2

Rewrite the expression.

$\frac{1}{2 \cos (θ)}$

Step 3.4

Separate fractions.

$\frac{1}{2} \cdot \frac{1}{\cos (θ)}$

Step 3.5

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

$\frac{1}{2} \sec (θ)$

Step 3.6

Combine $\frac{1}{2}$ and $\sec (θ)$ .

$\frac{\sec (θ)}{2}$

Step 4

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

$\frac{\cos (\frac{π}{2} - θ)}{\sin (2 θ)} = \frac{\sec (θ)}{2}$ is an identity