Question

Find the value of \( \sin [\cot^{-1} \sqrt{2} (\cos (\tan^{-1} 1))] \).

Hint:

Always work composition problems from the inside out. Simplifying the inner parts often converts complicated radical terms back into standard key trigonometric angles.

Answer 1

Concept: This problem requires evaluating a nested composition of trigonometric and inverse trigonometric functions from the innermost parentheses working outward step-by-step.

Step 1: Evaluate the innermost inverse trigonometric function.
The innermost expression is \( \tan^{-1}(1) \). We know that: \[ \tan\left(\frac{\pi}{4}\right) = 1 \quad \Rightarrow \quad \tan^{-1}(1) = \frac{\pi}{4} \]

Step 2: Move to the next outer layer involving cosine.
Substitute \( \frac{\pi}{4} \) back into the expression: \[ \cos(\tan^{-1} 1) = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

Step 3: Multiply by the coefficient inside the cotangent inverse.
Now evaluate the full expression inside \( \cot^{-1} \): \[ \sqrt{2} \cdot \cos(\tan^{-1} 1) = \sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1 \] So the overall expression reduces down to: \[ \sin [ \cot^{-1}(1) ] \]

Step 4: Evaluate the outer trigonometric operations.
We know that \( \cot^{-1}(1) = \frac{\pi}{4} \) because \( \cot\left(\frac{\pi}{4}\right) = 1 \). Substituting this value back into the expression: \[ \sin [ \cot^{-1}(1) ] = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \] Wait, let us carefully re-read the printed image expression text: The text reads \( \cot^{-1} \sqrt{2} (\cos (\tan^{-1} 1)) \). Let's re-verify if \( \sqrt{2} \) multiplies the whole cosine function. Yes, our step 3 calculation yields 1, and \( \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \).