Consider the function $$f(\theta ) = 4({\sin ^2}\theta + {\cos ^4}\theta )$$

Consider the following statements

1. $$f(\theta ) = 2$$ has no solution.

2. $$f(\theta ) = {7 \over 2}$$ has a solution.

Which of the above statement(s) is/are correct?

Consider a $$\Delta$$ABC in which $$\cos A + \cos B + \cos C = \sqrt 3 \sin {\pi \over 3}$$

What is the value of $$\sin {A \over 2}\sin {B \over 2}\sin {C \over 2}$$ ?

Consider a $$\Delta$$ABC in which $$\cos A + \cos B + \cos C = \sqrt 3 \sin {\pi \over 3}$$

What is the value of $$\cos \left( {{{A + B} \over 2}} \right)\cos \left( {{{B + C} \over 2}} \right)\cos \left( {{{C + A} \over 2}} \right)$$ ?

Given that tan$$\alpha$$ and tan$$\beta$$ are the roots of the equation x² + bx + c = 0 with b $$\ne$$ 0.

What is tan ($$\alpha$$ + $$\beta$$) equal to?