Question

Prove that cos6x+6cos4x+15cos2x+10/cos5x+5cos3x+10cosx= 2cosx

Answer 1

L.H.S = cos 6x + 6 cos  4x  + 15 cos  2x + 10/ cos 5x + 5 cos 3x + 10 cos x

= cos 6x + cos 4x + 5 cos 4x + 5 cos 2x + 10 cos 2x + 10/ cos 5x + 5 cos 3 x + 10 cosx

as cos 6x + cos 4x = 2 cos 5x cosx ; cos 4 x + cos 2x = 2 cos 3 xcosx; cos  2 x + cos 0 = 2 cosxcosx

= 2 cos 5xcosx + 5 (2 cos 3 xcosx) + 10 (2 cosxcosx)/ cos 5x + 5 cos  3 x + 10 cos x

= 2 cos x (cos 5x + 5 cos  3x + 10 cosx)/ cos 5x + 5cos 3x + 10 cosx

= 2 cosx

= R.H.S

Answer 2

Given,

cos6x+6cos4x+15cos2x+10/cos5x+5cos3x+10cosx= 2cosx

L.H.S = cos 6x + 6 cos  4x  + 15 cos  2x + 10/ cos 5x + 5 cos 3x + 10 cos x

= cos 6x + (cos 4x + 5 cos 4x) + (5 cos 2x + 10 cos 2x) + 10/ cos 5x + 5 cos 3 x + 10 cosx

= 2 cos 5xcosx + 5 (2 cos 3 xcosx) + 10 (2 cosxcosx)/ cos 5x + 5 cos  3 x + 10 cos x

= 2 cos x (cos 5x + 5 cos  3x + 10 cosx)/ cos 5x + 5cos 3x + 10 cosx

= 2 cosx

= R.H.S