Evaluate the determinants in Exercises 1 and 2. 1. 2 41 1. 5 1 2. (1) cos 0 sin e -sin 0 cos x? - x + 1 x -1 x + 1 x +1

Evaluate: \int_{0}^(pi) sin x (1+ 2cos x)(1+cos x)^2 dx

{ cos (sin ^(-1) x)}^(2) ={sin (cos ^(-1) x)}^(2)

Evaluate: cos (2 cos ^(-1) x +sin ^(-1) x ) at x = (1)/(5).

Evaluate: cos (2 cos ^(-1) x +sin ^(-1) x ) at x = (1)/(5).

If x , y in R , then the determinant = |(cos x , -sin x,1),(sin x, cos x,1),(cos(x+y), -sin(x+y), 0)| lies in the interval (a) [-sqrt(2),sqrt(2)] (b) [-1,1] (c) [-sqrt(2),1] (d) [-1,-sqrt(2)]

if a=(1+sin x)/(1-cos x+sin x) then (1+cos x+sin x)/(2sin x) is

int_(0)^(pi//2) (sin x cos x) / ((1 + 2 sin x ) ( 1+ sin x))dx =

1- (sin ^ (2) x) / (1 + cos x) + (1 + cos x) / (sin x) - (sin x) / (1-cos x) =

Find the value of sin ( 2 cos^(-1) x + sin^(-1) x) " when " x = 1/5