Question
Find the value of:
\[ \sin(\cot^{-1}x) \]
Solution
Let
\[ \theta = \cot^{-1}x \]
Then,
\[ \cot \theta = x = \frac{\text{Adjacent}}{\text{Opposite}} \]
Take a right triangle such that:
- Adjacent = \( x \)
- Opposite = \( 1 \)
Then hypotenuse:
\[ \sqrt{x^2 + 1} \]
Now,
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{\sqrt{1 + x^2}} \]
Therefore,
Final Answer:
\[ \boxed{ \sin(\cot^{-1}x) = \frac{1}{\sqrt{1 + x^2}} } \]
Key Concept
Using triangle representation is the easiest way to convert inverse trigonometric expressions into algebraic form.