Question

FX equal to X cosec x

Answer 1

Let g(x) = sin x and h(x) = 1 + cos x, function f may be considered as the quotient of functions g and h: f(x) = g(x) / h(x). Hence we use the quotient rule, f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x) 2, to differentiate function f as follows 

g '(x) = cos x 

h '(x) = - sin x 

f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x)2

= [ (1 + cos x)(cos x) - (sin x)(- sin x) ] / (1 + cos x) 2

= [ cos x + cos 2x + sin 2x ] / (1 + cos x) 2

Use trigonometric identity cos 2x + sin 2x = 1 to simplify the above 

f '(x) = [ cos x + 1 ] / (1 + cos x) 2 = 1 / [cos x + 1]