Animated Solution for Mathematics - Differentiation: If $y(\alpha )=\sqrt{2(\frac{\tan \alpha +\cot \alpha }{1+{\tan }^2\alpha })+\frac{1}{{\sin }^2\alpha }}$ where $\alpha \in (\frac{3\pi }{4},\pi )$, then $\frac{dy}{d\alpha }$ at $\alpha =\frac{5\pi }{6}$ is

Imagine you are standing on the unit circle, looking at the expression:

It looks like a terrifying, tangled mess of trigonometric functions. The secret to solving these problems is not brute force, but elegant simplification.

Let's start with the denominator inside the square root: $1+{\tan }^2\alpha$. This is a classic identity:

Now, look at the numerator: $\tan \alpha +\cot \alpha$. If we convert these to sines and cosines, we get:

When we divide this by ${\sec }^2\alpha$ (which is $\frac{1}{{\cos }^2\alpha }$), the expression simplifies beautifully:

The monster is already shrinking!

Now, substitute this back into our main equation:

We know $\frac{1}{{\sin }^2\alpha }={\csc }^2\alpha$. Using the identity ${\csc }^2\alpha =1+{\cot }^2\alpha$, we obtain:

Do you see it? This is a perfect square:

This is where most students stumble. The square root of a square is the absolute value:

We must respect the interval $\alpha \in (\frac{3\pi }{4},\pi )$. In this part of the second quadrant, $\cot \alpha <-1$.

Therefore, $1+\cot \alpha$ is negative. To remove the modulus, we must negate the expression:

Now, the differentiation is a breeze:

Finally, we evaluate this at $\alpha =\frac{5\pi }{6}$. We know $\csc (\frac{5\pi }{6})=2$, so:

We have conquered the monster! The final answer is 4.