If y² + log_e (cos²x) = y,
$$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$, then :

Let ƒ and g be differentiable functions on R such that fog is the identity function. If for some a, b $$ \in $$ R, g'(a) = 5 and g(a) = b, then ƒ'(b) is equal to :

If $$x = 2\sin \theta - \sin 2\theta $$ and $$y = 2\cos \theta - \cos 2\theta $$,
$$\theta \in \left[ {0,2\pi } \right]$$, then $${{{d^2}y} \over {d{x^2}}}$$ at $$\theta $$ = $$\pi $$ is :

Let ƒ(x) = (sin(tan^–1x) + sin(cot^–1x))² – 1, |x| > 1.
If $${{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\sin }^{ - 1}}\left( {f\left( x \right)} \right)} \right)$$ and $$y\left( {\sqrt 3 } \right) = {\pi \over 6}$$, then y($${ - \sqrt 3 }$$) is equal to :