" 5.(i) "cos x=(-1)/(2)

Prove that (i).(sin 2x)/(1-cos 2x) = cot x" " (ii) .(1- cos 2x)/(1+ cos 2x) = tan^(2) x

(1)/(2)cos^(-1)((5cos x+3)/(5+3cos x))=tan^(-1)((1)/(2)(tan x)/(2))

cos^(-1) ""(5)/(x)+cos^(-1) ""(12)/(x)=(pi)/(2)

Evaluate: (i) int(x^(3)-3x^(2)+5x-7+x^(2)a^(x))/(2x^(2))dx (ii) int(cos x)/(1+cos x)dx

Evaluate: (i) int(1-cos x)/(1+cos x)dx (ii) int{3sin x-4cos x+(5)/(cos^(2)x)-(6)/(sin^(2)x)+tan^(2)x-cot^(2)x}dx

If I_(n)=int cos^(n)x dx , prove that, I_(n)=(1)/(n)cos^(n-1)x sin x+(n-1)/(n).I_(n-2) .Hence, evaluate, int cos^(5)x dx .

Let f(x)=(2cos x-1)(2cos2x-1)(2cos2^(2)x-1)...(2cos2^(n)x-1) (where n>=1). Then prove that f((2 pi k)/(2^(n)+-1))=1AA k in I

Evaluate: (i) int(cos2x+x+1)/(x^(2)+sin2x+2x)dx (ii) int(1)/(cos(x+a)cos(x+b))dx

If y=cos^(-1)(cos x) then (dy)/(dx) at x=(5 pi)/(4) is (i) 1( ii) -1( iii) (1)/(sqrt(2))( iv) sqrt(2) at x=(5 pi)/(4) is (i)

Prove that ""(1)/(2)cos^(-1)""((5cos x+3)/(5+3 cos x))= tan^(-1)""((1)/(2)tan""(x)/(2))