Question

If \( 2\cos^2 45^\circ - 1 = \cos \theta \), then find the value of \( \theta \).

Hint:

Use standard trigonometric values: \( \cos 45^\circ = \frac{1}{\sqrt{2}} \), \( \sin 90^\circ = 1 \), and \( \cos 90^\circ = 0. \)

Answer 1

Step 1: Simplify the left-hand side.
\[ 2\cos^2 45^\circ - 1 = 2\left(\frac{1}{\sqrt{2}}\right)^2 - 1 = 2\left(\frac{1}{2}\right) - 1 = 1 - 1 = 0 \]
Step 2: Substitute in equation.
\[ \cos \theta = 0 \]
Step 3: Find the value of \( \theta \).
\[ \theta = 90^\circ \]
Step 4: Final Answer.
\[ \boxed{\theta = 90^\circ} \]