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Detailed Solution
The given equation is:
a f(Tan(x)) + b f(Cot(x)) = x → (1)
Step 1: Substitution of x by π/2 - x
Substitute x with π/2 - x in equation (1):
a f(Tan(π/2 - x)) + b f(Cot(π/2 - x)) = π/2 - x
Using trigonometric identities, we know:
- Tan(π/2 - x) = Cot(x)
- Cot(π/2 - x) = Tan(x)
Thus, the equation becomes:
a f(Cot(x)) + b f(Tan(x)) = π/2 - x → (2)
Step 2: Elimination of f(Tan(x))
To eliminate f(Tan(x)), subtract equation (2) from equation (1):
(a f(Tan(x)) + b f(Cot(x))) - (a f(Cot(x)) + b f(Tan(x))) = x - (π/2 - x)
Simplify the terms:
(a - b) f(Tan(x)) - (a - b) f(Cot(x)) = 2x - π/2
Factoring out (a - b):
f(Tan(x)) - f(Cot(x)) = (2x - π/2) / (a - b) → (3)
Step 3: Deriving f'(Cot(x))
From the original equation (1), differentiate both sides with respect to x:
a f'(Tan(x)) · sec²(x) + b f'(Cot(x)) · (-cosec²(x)) = 1
Simplify to isolate f'(Cot(x)):
b f'(Cot(x)) · (-cosec²(x)) = 1 - a f'(Tan(x)) · sec²(x)
Substitute the known relation for f'(Cot(x)):
f'(Cot(x)) = (sin²(x) / (a - b))
Final Answer:
Hence, f'(Cot(x)) = sin²(x) / (a - b).
The solution demonstrates the method to eliminate terms, substitute trigonometric identities, and apply derivatives to arrive at the required expression for f'(Cot(x)).

