Show that: `sin(pi/2 - x)` = cos x`

Solve the equation ( sin x + cos x )^(1+ sin 2x )=2 , when -pi le x le pi .

Show that (i) int_(0)^(pi//2)f(sinx) d x=int_(0)^(pi//2)f(cos x) d x (ii) int_(0)^(pi//2)f(tan x) d x=int_(0)^(pi//2)f(cot x) d x (iii) int_(0)^(pi//2)f(sin 2 x) sin xd x = int_(o)^(pi//2)f(sin 2x).cosx d x

Prove that : int_(0)^(pi//2) (cos^(5))/(sin^(5) x+cos^(5) x)dx= (pi)/(4)

If|a|lt 1, show that int _(0)^(pi)(log(1+a cos x ))/( cos x)dx =pi sin^(-1) a

Show that '(1-cos 2x + sin x)/(sin 2x + cos x) = tan x'

The solutions of the equation 1+(sin x - cos x)"sin" pi/4=2 "cos"^(2) (5 x)/2 is/are

Prove that: cos(pi/4-x)cos(pi/4-y)-sin(pi/4-x)sin(pi/4-y)=sin(x+y)

Prove that sin^(-1) cos (sin^(-1) x) + cos^(-1) x) = (pi)/(2), |x| le 1

Number of solutions of the equation sin x + cos x-2sqrt(2) sin x cos x=0 for x in [0, pi] is

Find the solution of the equation ( sin x + cos x ) sin 2 x =a(sin^(3) x + cos^(3) x) located between (pi)/(2) and pi and for which values of 'a' does this equation have at most one solution satisfying the condition (pi)/(2) le x le pi .