Question

Prove that cotx cot2x-cot2x cot3x-cot3x cotx=1

Answer 1

L.H.S=cot x. cot 2x - cot 2x. cot3x - cot x. cot3x

=cot x. cot 2x - cot3x(cot 2x +cot x)

=cotx .cot 2x - cot (x+2x) .(cot 2x+ cot x) [we can write cot3x as cot(x+2x)]

=cotx.cot2x- {(cot x. cot 2x -1)/(cot x + cot 2x)}.(cot x + cot 2x) [since cot(a+b)=(cot a. cot b -1)/(cot b+cot a)]

=cot x. cot 2x -cot x. cot 2x +1

=1

=R.H.S

Answer 2

Answer:1

Step-by-step explanation:cotx. cot2x-cot2x.cot3x-cot3x.cosx

=cosx.cos2x/sinx.sin2x-cos2x.cos (2x+x)/sin2x.sin (2x+x)-cos (2x+x)cosx/sin(2x+x)sinx

=cos^2 x.cos2x.sin2x+cos^2 2x.sinx.cosx-cos^2 2x.cosx.sinx+cos2x.sin2x.sin^2 x-cos^2 x.cos2x.sin2x+cosx.sin^2 2x.sinx/sinx.sin2x (sin2x.cosx+cos2x.sinx)

=sin2x.sinx (cos2x.sinx+cosx.sin2x)/sinx.sin2x (sin2x.cosx+cos2x.sinx)

=1