Question

Given that \( \sin x = \cos^2 x \), find the value of \( \frac{\cos x}{\sin^2 x} \).

Hint:

In trigonometric problems, manipulating the given identities can often simplify the expression and lead to the final solution.

Answer 1

Step 1: Using the given identity.
We are given that \( \sin x = \cos^2 x \). This can be rewritten as: \[ \sin x = \left( \cos x \right)^2 \]
Step 2: Solve for \( \frac{\cos x}{\sin^2 x} \).
We need to find the value of \( \frac{\cos x}{\sin^2 x} \). Substituting \( \sin x = \cos^2 x \) into the expression: \[ \frac{\cos x}{\sin^2 x} = \frac{\cos x}{\left( \cos^2 x \right)^2} = \frac{\cos x}{\cos^4 x} = \frac{1}{\cos^3 x} \]
Step 3: Simplify further.
After solving the trigonometric equation, we get: \[ \frac{\cos x}{\sin^2 x} = \left( \frac{\sqrt{5}+1}{2} \right)^{3/2} \]