`(cos (pi +x) cos (-x))/( sin (pi -x) cos ((pi )/(2) + x))= cot ^(2) x`

Prove that cos ( ( 3pi )/(2) + x) ) cos (2pi+x) .[ cot ((3pi )/( 2) - x) ) + cot (2pi +x) ]=1

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

cos ((3pi)/( 4) + x) - cos((3pi)/( 4) -x) =- sqrt2 sin x

If cos ^(-1) x+sin ^(-1) (x)/(2)=(pi)/(6) then x=

Prove that cos (pi/2 + x) = - sin x .

Prove geometrically that cos(x + y) = cos x cos y - sinx sin y and hence prove cos((pi)/(2) + x) = - sin x .

sin ^(-1) x=(pi)/(5) , then cos ^(-1) x=

If x ne n pi, x ne (2n+1) (pi)/(2) n in Z, then : sin^(-1) ((sin^(-1) (cos x) + cos ^(-1) (sin x))/(tan^(-1) (cot x) + cot^(-1) (tan x )))=