Animated Solution for Mathematics - Inverse Trigonometric Functions: Prove that $\cos {\tan }^{-1}\sin {\cot }^{-1}x=\sqrt{\frac{x^2+1}{x^2+2}}$.

Imagine you are standing before a complex, nested structure: $\cos ({\tan }^{-1}(\sin ({\cot }^{-1}x)))$. It looks like a labyrinth, but the secret is to peel it layer by layer.

Let us embark on this journey together by resolving the innermost core first.

The Innermost Core

We begin at the heart of the expression: ${\cot }^{-1}x$. Let us define this as an angle, $\theta$.

Imagine a right-angled triangle where the angle is $\theta$. Since $\cot \theta$ is the ratio of the base to the perpendicular, our base is $x$ and our perpendicular is $1$.

Using the Pythagorean theorem, the hypotenuse is $\sqrt{x^2+1}$. Now, we find the sine of this angle:

The Bridge

Substitute this result back into our original expression. The expression $\cos ({\tan }^{-1}(\sin ({\cot }^{-1}x)))$ transforms into:

Let us define this new inner term as a new angle, $\varphi$:

The Final Reveal

We construct our second right-angled triangle for angle $\varphi$. The perpendicular is $1$, and the base is $\sqrt{x^2+1}$.

To find the hypotenuse, we apply the Pythagorean theorem again:

Finally, we need the cosine of this angle $\varphi$. From our second triangle:

Combining these under a single radical, we reach the final result:

The labyrinth is solved. By breaking the problem down into simple, geometric steps, we have arrived at the proof.