52.(1-cos^2theta)(1+cot^2theta)
=sin^2theta*csc^2theta
=1
53.cos(2x-3y)
=cos2xcos3y+sin2xsin3y
54.sin2xcosx-cos2xsinx
sin(2x-x)=sinx
55.(1-cos2x)/(sin2x)
=(2sin^2x)/(2sinxcosx)
=sinx/cosx=tanx
56.cos4x
=2cos^2 2x-1
=2(2cos^2x-1)^2-1
=4cos^4x-8cos^2x+2-1
:.cos4x=4cos^4x-8cos^2x+1..(1)
Given
cos4x=pcos^4x+qcos^x+r...(2)
Comparing (1) and (2)
we have
p=4,q=-8,r=1
57.
costheta/(1-sintheta)
=(cos^2(theta/2)-sin^2(theta/2))/(cos^2(theta/2)+sin^2(theta/2)-2sin(theta/2)cos(theta/2))
=(cos^2(theta/2)-sin^2(theta/2))/(cos(theta/2)-sin(theta/2))^2
=(cos(theta/2)+sin(theta/2))/(cos(theta/2)-sin(theta/2))
Dividing numerator and denominator by cos(theta/2)
we get
=(1+tan(theta/2))/(1-tan(theta/2))
=(t+1)/(1-t)
Putting tan(theta/2)=t
