Question

The value of \[ \tan^{-1} \left( \frac{\sin 2 - 1}{\cos 2} \right) \] is:

• A. \( 1 - \frac{\pi}{4} \)
• B. \( \frac{\pi}{2} - 1 \)
• C. \( 2 - \frac{\pi}{2} \)
• D. \( \frac{\pi}{4} - 1 \)

Hint:

When dealing with inverse trigonometric functions, simplify the expression using standard trigonometric identities before calculating the inverse.

Answer 1

The Correct Option is A

Step 1:
We start with the expression \( \tan^{-1} \left( \frac{\sin 2x - 1}{\cos 2x} \right) \). We simplify this expression using trigonometric identities. \[ \frac{\sin 2x - 1}{\cos 2x} = \frac{-(1 - \sin 2x)}{\cos 2x} \] Using the standard trigonometric identity, this simplifies to: \[ \tan^{-1} \left( \frac{-(1 - \sin 2x)}{\cos 2x} \right) \]
Step 2:
The solution to this inverse tangent function simplifies to the form \( 1 - \frac{\pi}{4} \).
Step 3:
Hence, the value of the
given expression is \( 1 - \frac{\pi}{4} \).