Prove that 2sin^2 y + 4cos(x + y) sinxsiny ny + cos 2 * (x + y) = cos 2x

Asked by apnp31 | 21st Sep, 2020, 12:19: PM

Expert Answer:

To prove: 2sin2y + 4cos(x+y) sinx siny + cos2(x+y) = cos2x

LHS equals 2 sin squared straight y plus 4 cos left parenthesis straight x plus straight y right parenthesis space sinx space siny space plus thin space cos left parenthesis 2 straight x plus 2 straight y right parenthesis
equals 2 sin squared straight y space plus space 4 open square brackets cosx space cosy space minus space sinx space siny close square brackets space sinx space siny space plus thin space cos 2 straight x space cos 2 straight y space minus space sin 2 straight x space sin 2 straight y
equals 2 sin squared straight y space plus space 4 cosx space cosy space sinx space siny space minus space 4 sin squared straight x space sin squared straight y space plus thin space cos 2 straight x space cos 2 straight y space minus space sin 2 straight x space sin 2 straight y
equals 2 sin squared straight y space plus open parentheses 2 cosx space sinx close parentheses space open parentheses 2 siny space cosy close parentheses space minus space 4 sin squared straight x space sin squared straight y space plus thin space cos 2 straight x space cos 2 straight y space minus space sin 2 straight x space sin 2 straight y
equals 2 sin squared straight y space plus space sin 2 straight x space sin 2 straight y space minus space 4 sin squared straight x space sin squared straight y space plus thin space cos 2 straight x space cos 2 straight y space minus space sin 2 straight x space sin 2 straight y
equals 2 sin squared straight y space minus space 4 sin squared straight x space sin squared straight y space plus thin space cos 2 straight x space cos 2 straight y
equals 2 sin squared straight y space open square brackets 1 minus 2 sin squared straight x close square brackets space plus thin space cos 2 straight x space cos 2 straight y
equals 2 sin squared straight y space cos 2 straight x space plus thin space cos 2 straight x space cos 2 straight y
equals cos 2 straight x open square brackets 2 sin squared straight y plus cos 2 straight y close square brackets
equals cos 2 straight x open square brackets 2 sin squared straight y plus 1 minus 2 sin squared straight y close square brackets
equals cos 2 straight x
Hence, proved.

Answered by Renu Varma | 22nd Sep, 2020, 01:37: PM