Question

prove that tanx +2tan2x +4tan4x +8cot8x = cotx

Answer 1

tanx +2tan2x+4tan4x+8cotx=cotx
tanx-cotx=tan^2x-1/tanx=1/tanx/(tan^2x-1)
=-2/2tanx/(1-tan^2x)
=-2/tan2x=-2cot2x
hence
tanx+2cot2x=cotx
hence use this concept here
tanx+2tan2x+4tan4x+8cot8x
=tanx+2tan2x+4 (tan4x+2cot8x)
=tanx+2tan2x+4cot4x
=tanx+2 (tan2x+2cot4x)
=tanx+2cot2x
=cotx=RHS

Answer 2

Sin 8x = 2sin(4x)cos(4x)

=4sin(2x)cos(2x)cos(4x)

=8sin(x)cos(x)cos(2x)cos(4x)

Now if we use logs,

log{sin(8x)}=log{8sin(x)} + log{cos(x)} + log{cos(2x)} + log{cos(4x)}

Diffenciate,

8cot(8x)=cotx - tanx - 2tan(2x) - 4tan(4x)

Now,Cotx=tanx + 2tan2x + 4tan4x + 8cot8x

Hope it was helpful... Please mark it as brainliest.