Determine if True (cos(pi/2-theta))/(sin(2theta))=(sec(theta))/2

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

Start on the left side.

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

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

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

Simplify the expression.

Simplify the numerator.

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

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

Multiply $0$ by $\cos (θ)$ .

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

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

$\frac{0 + 1 \sin (θ)}{\sin (2 θ)}$

Multiply $\sin (θ)$ by $1$ .

$\frac{0 + \sin (θ)}{\sin (2 θ)}$

Add $0$ and $\sin (θ)$ .

$\frac{\sin (θ)}{\sin (2 θ)}$

Apply the sine double-angle identity.

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{2 \cos (θ)}$

Separate fractions.

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

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

$\frac{1}{2} \sec (θ)$

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

$\frac{\sec (θ)}{2}$

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