If y( α ) = √(2( (tan α + cot α)/(1 + tan ²α) ) + 1/(sin ²α)) ,α ∈ ( 3π/4,π )
dy/dα at α = 5π/6is :

Solution

Maths sir · 4 steps

The thing under the root is a perfect square, so watch the sign of 1 + cot α before you differentiate.

1. Simplify the fraction. tan α + cot α = 1/(sin α cos α) and 1 + tan² α = sec² α. Dividing gives (cos² α)/(sin α cos α) = cot α.
2. Spot the square. Write 1/(sin² α) = csc² α = 1 + cot² α. The expression under the root becomes 1 + 2 cot α + cot² α.

   y = √(1 + cot α)² = |1 + cot α|

3. Remove the modulus. On (3π/4, π), cot α falls from -1 towards -∞, so cot α < -1 and 1 + cot α < 0. Hence y = -1 - cot α.
4. So. dy/dα = csc² α. At α = 5π/6, sin α = 1/2, so csc² α = 4. The answer is 4, option (a).

The idea to keep. Whenever you see √(…)², it is a modulus. Use the given interval to decide its sign first, and only then differentiate.

Answer from the NTA final answer key. Solution by PrepWiser.