Animated Solution for Mathematics - Inverse Trigonometric Functions: $\text{cosec}\left[2{\cot }^{-1}(5)+{\cos }^{-1}\left(\frac{4}{5}\right)\right]$ is equal to:

Welcome, aspiring engineers! Today, we are going to dismantle a problem that, at first glance, looks like a tangled mess of inverse trigonometric functions. You might see $\text{cosec}\left[2{\cot }^{-1}(5)+{\cos }^{-1}\left(\frac{4}{5}\right)\right]$ and feel a moment of hesitation.

That is perfectly normal. But remember, every complex mathematical structure is built from simple, elegant bricks. Our mission is to identify those bricks and assemble them into a clear, logical path.

Deconstructing the Beast

Let us start by isolating the components. We have two distinct angles hidden inside that cosecant function. Let us define them clearly.

For the first term, let $\alpha ={\cot }^{-1}(5)$. This is a powerful substitution. It immediately tells us that $\cot \alpha =5$, which implies $\tan \alpha =\frac{1}{5}$.

Now, our expression contains $2\alpha$. We need to find the properties of this double angle. We reach into our toolkit and pull out the double angle identity for tangent:

Substituting our value, we get:

Simplifying this, we find $\tan (2\alpha )=\frac{2}{5}\times \frac{25}{24}=\frac{5}{12}$. Let us call this angle $\theta =2\alpha$. We now know $\tan \theta =\frac{5}{12}$.

The Power of Triangles

Now, let us visualize this. Imagine a right-angled triangle where the angle is $\theta$. Since $\tan \theta =\frac{\text{opposite}}{\text{adjacent}}=\frac{5}{12}$, we can construct a triangle with an opposite side of $5$ and an adjacent side of $12$.

By the Pythagorean theorem, the hypotenuse is:

From this triangle, we can instantly read off $\sin \theta =\frac{5}{13}$ and $\cos \theta =\frac{12}{13}$.

Now, let us turn to the second term: $\varphi ={\cos }^{-1}(\frac{4}{5})$. This implies $\cos \varphi =\frac{4}{5}$. Again, visualize a right-angled triangle. The adjacent side is $4$ and the hypotenuse is $5$. This is the classic $(3,4,5)$ triangle, so the opposite side must be $3$. Thus, $\sin \varphi =\frac{3}{5}$.

The Grand Synthesis

We have arrived at the final stage. Our original expression is now simply $\text{cosec}(\theta +\varphi )$. We know that $\text{cosec}(\theta +\varphi )=\frac{1}{\sin (\theta +\varphi )}$.

We use the compound angle formula:

Substituting our values:

Finally, we take the reciprocal to find the cosecant:

Look at that! The complexity has vanished, leaving behind a clean, beautiful fraction. You have successfully navigated the labyrinth. The final answer is $\frac{65}{56}$.