Find the amplitude of `z= sin alpha + i(1-cos alpha), 0 lt alpha lt pi`.

If sinalpha=15/17 , cosbeta=12/13 and alpha and beta are in the first quadrant , find the values of sin(alpha+beta) , cos(alpha+beta) , and tan(alpha+beta)

Find the integral (cos 2 x-cos 2 alpha)/(cos x-cos alpha)

If the equation 3x + 3y + 5=0 is written in the form x cos alpha + y sin alpha = p , then the value of sin alpha + cos alpha is a) sqrt2 b) (1)/( sqrt2) c) - sqrt2 d) - (1)/( sqrt2)

Find inverse of the matrix A=[[cos alpha, -sin alpha, 0 ],[ sin alpha, cos alpha, 0],[ 0, 0, 1]]

Let lambda and alpha be real. Let S denote the set of all values of lambda for which the system of linear equations lambda x+ (sin alpha) y + (cos alpha)z = 0 , x + (cos alpha) y + (sin alpha) z = 0 , -x + (sin alpha) y - (cos alpha) z = 0 has a non trivial solution then S contains a) (-1,1) b) [-sqrt2,-1] c) [1,sqrt2] d) [-sqrt2,sqrt2]

If u = cot^(-1)sqrt(tan alpha ) - tan ^(-1)sqrt(tan alpha) then tan (pi/4-u/2) is a) sqrt(tan alpha) b) sqrt(cot alpha) c) tan alpha d) cot alpha