Question

Evaluate \( \tan \left[ \cos^{-1} \left( \tan \frac{3\pi}{4} \right) \right] \).

Hint:

Always work from the inside out in nested trigonometric expressions. Ensure results of inverse functions lie within their defined principal value branches to avoid errors.

Answer 1

Concept:
• Evaluate the trigonometric expression starting from the innermost function and moving outwards.
• Use standard values for trigonometric functions of specific angles.
• Principal value branch for \( \cos^{-1}x \) is \( [0, \pi] \).

Step 1: Evaluate the innermost tangent function
The innermost term is \( \tan \left( \frac{3\pi}{4} \right) \).
We can write \( \frac{3\pi}{4} \) as \( \pi - \frac{\pi}{4} \).
\[ \tan \left( \pi - \frac{\pi}{4} \right) = -\tan \left( \frac{\pi}{4} \right) = -1 \]

Step 2: Evaluate the inverse cosine function
Substitute the result from Step 1 into the inverse cosine function:
\[ \cos^{-1}(-1) \]
In the principal branch \( [0, \pi] \), the cosine function is \( -1 \) at \( \pi \).
\[ \cos^{-1}(-1) = \pi \]

Step 3: Evaluate the final tangent function
Substitute the result from Step 2 into the outer tangent function:
\[ \tan(\pi) \]
Since \( \sin(\pi) = 0 \) and \( \cos(\pi) = -1 \), we have:
\[ \tan(\pi) = \frac{0}{-1} = 0 \]