Question

If
$ \sec(5^\circ - 2\theta) = \csc(5\theta - 5^\circ),\ 0^\circ < \theta < 90^\circ $,
then what will be the value of 
$ \frac{\sin 3\theta + \tan 2\theta}{\sec 2\theta} $?

• A. \( \frac{3\sqrt{3}}{4} \)
• B. \( \frac{3\sqrt{3}}{2} \)
• C. \( \frac{2 + \sqrt{3}}{2} \)
• D. \( \frac{1 + \sqrt{3}}{2} \)

Hint:

When solving trigonometric equations, it's helpful to use identities such as \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \) to transform the equation into a more manageable form.

Answer 1

The Correct Option is D

We are given the equation \[ \sec(5^\circ - 2\theta) = \csc(5\theta - 5^\circ). \] Now, use the identity for secant and cosecant: \[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \csc x = \frac{1}{\sin x}. \] Substitute these into the equation: \[ \frac{1}{\cos(5^\circ - 2\theta)} = \frac{1}{\sin(5\theta - 5^\circ)}. \] This implies: \[ \cos(5^\circ - 2\theta) = \sin(5\theta - 5^\circ). \] We know the identity \( \cos x = \sin(90^\circ - x) \). Thus, we can write: \[ 5^\circ - 2\theta = 90^\circ - (5\theta - 5^\circ). \] Simplifying this equation: \[ 5^\circ - 2\theta = 90^\circ - 5\theta + 5^\circ. \] \[ 5^\circ - 2\theta = 95^\circ - 5\theta. \] \[ 3\theta = 90^\circ. \] \[ \theta = 30^\circ. \] Now, we need to calculate the value of \( \frac{\sin 3\theta + \tan 2\theta}{\sec 2\theta} \) for \( \theta = 30^\circ \). Substitute \( \theta = 30^\circ \): \[ \sin(3 \times 30^\circ) = \sin 90^\circ = 1, \] \[ \tan(2 \times 30^\circ) = \tan 60^\circ = \sqrt{3}, \] \[ \sec(2 \times 30^\circ) = \sec 60^\circ = \frac{2}{\sqrt{3}}. \] Now, substitute these values into the given expression: \[ \frac{\sin 3\theta + \tan 2\theta}{\sec 2\theta} = \frac{1 + \sqrt{3}}{\frac{2}{\sqrt{3}}} = \frac{(1 + \sqrt{3}) \cdot \sqrt{3}}{2}. \] Simplify the numerator: \[ = \frac{\sqrt{3} + 3}{2}. \] Thus, the value is \( \frac{1 + \sqrt{3}}{2} \).