Prove that for any positive integer `k ,(sin2k x)/(sinx)=2[cosx+cos3x++cos(2k-1)x]dot` Hence, prove that `int_0^(pi/2)sin2x kcotxdx=pi/2dot`

Prove that for any positive integer K, (sin2kx)/(sinx)=2[cos x + cos 3x+....+ cos (2k-1)x] Hence , proved that int_(0)^(pi//2) sin 2kx. Cot x dx = (pi//2)

(sin 2 kx)/(sin x)=2[cos x + cos 3 x + …+cos (2k -1)x] , then value of I=int_(0)^(pi//2)(sin 2 k x)/(sin x)cos x dx is :

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

If for k in N backslash(sin2kx)/(sin x)=2[cos x+cos3x+...+cos(2k-1)x] Then the value of I=int_(0)^( pi/2)sin2kx*cot x backslash dx is

int(sin2x-sin2k)/(sin x-sin k+cos x-cos k)dx

Prove that : int_(0)^(pi//2) (sin x-cos x)/(1+sin x cos x)dx=0 " (ii) Prove that " : int_(0)^(pi//2) sin 2x. log (tan-x) dx=0

Prove that cosx+cos2x+ … + cosnx=(sin(n+1/2)x-sin(x/2))/(2sin(x/2)) and hence prove that : int_0^1(sin(n+1/2)x)/sin(x/2)dx=pi

Prove that int_(0)^((pi)/(2))(sin^(2)x)/(1+sin x cos x)dx=(pi)/(3sqrt(3))

int_(0)^(pi//2) (sinx )/(sin x + cos x ) dx=

Prove that the integral int_(0)^(pi) (sin 2 k x)/(sin x) dx equals zero if k an integer.