Question

If f(x) = `(sin^2x)/(1 + cotx) + (cos^2x)/(1 + tan x)`, then `"f'"(pi/4)` is ______.

Options

• `sqrt(3)`

• `1/sqrt(3)`

• 0

• `-sqrt(3)`

Answer 1

If f(x) = `(sin^2x)/(1 + cotx) + (cos^2x)/(1 + tan x)`, then `"f'"(pi/4)` is 0.

Explanation:

f(x) = `(sin^2x)/(1 + cotx) + (cos^2x)/(1 + tan x)`

= `(sin^2x(sin x))/(sin x + cos x) + (cos^2x(cosx))/(cosx + sinx)`

=`(sin^3x + cos^3x)/(sinx + cosx)`

= `sin^2x - sinx cosx + cos^2x`  ......[∵ a³ + b³ = (a + b)(a² – ab + b²)]

= `(sin^2x + cos^2x) - 1/2 (2 sin x cosx)`

= `1 - 1/2 sin 2x`

∴ f'(x) = – cos 2x

⇒ `"f'"(pi/4) = -cos(pi/2)` = 0