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Question Number 16085 by Tinkutara last updated on 18/Jun/17

Let f (\sin x) + 2 f (\cos x) = 3 \forall x \in (0, \frac{π}{2}) .

Then

(1) f (\sin x) = 1, x \in (0, \frac{π}{2})

(2) f (\sin x) = 1, x \in (- 1, 0)

(3) f (\cos x) = 1, x \in (0, 1)

(4) f (x) = 1, x \in (0, 1)

Commented by prakash jain last updated on 19/Jun/17

f (\sin x) + 2 f (\cos x) = 3

x = \frac{π}{2} - x

f (\cos x) + 2 f (\sin x) = 3

f (\cos x) - f (\sin x) =

\Rightarrow f (\cos x) = f (\sin x)

\Rightarrow 3 f (\cos x) = 3 \Rightarrow f (\cos x) = 1 = f (\sin x)

f (\sin x) = 1 = f (\cos x), x \in (0, \frac{π}{2})

\Rightarrow f (x) = 1, x \in (0, 1)