Question

The value of \( \tan\!\left\{ \cos^{-1}\!\left(\frac{\sqrt{2}}{2}\right) - \frac{\pi}{2} \right\} \) is __________.

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

Hint:

Always evaluate inverse trigonometric values first, then simplify angles before applying trigonometric identities.

Answer 1

The Correct Option is A

Step 1: Evaluate inverse cosine.
\[ \cos^{-1}\left(\frac{\sqrt{2}}{2}\right)=\frac{\pi}{4} \]

Step 2: Substitute into expression.
\[ \tan\left(\frac{\pi}{4}-\frac{\pi}{2}\right) \]

Step 3: Simplify angle.
\[ \frac{\pi}{4}-\frac{\pi}{2} = -\frac{\pi}{4} \]

Step 4: Use identity.
\[ \tan(-\theta) = -\tan\theta \]

Step 5: Apply identity.
\[ \tan\left(-\frac{\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) \]

Step 6: Evaluate tangent.
\[ \tan\left(\frac{\pi}{4}\right)=1 \]
\[ \Rightarrow -1 \]

Step 7: Final conclusion.
\[ \boxed{-1} \]