Prove that: tan4x= cos8x+cos3x/sin8x+sin3x

Question

This question is about to prove some trigonometric identities

Answer 1

tan4x= (8cosx + 3cosx)/(8sinx +3sinx)

1st see if it's true for a couple angles x

such as x= 0

tan4(0) = tan0 = 0

while Right Side is undefined, that's not promising for an "Identitty" true for all x

actually that should end it, as proof is impossible

but try another x value = pi

tan4pi = 0

same problem

try x=pi/2

tan 4(pi/2) = tan2pi = 0

RS = (0)/(sin8pi/2 +sin3pi/2) = 0/(0) = undefined

try x= pi/4 = 45 degrees

tan4x = tanpi = 0

RS = (cos2pi + cos3pi/4)/(sin2pi + sin3pi/4) = (1 + 1/sqr2)/(0+1/sqr2) =sqr2 +1 doesn't = 0

it's Not an identity, so no proof is possible DNE, Null Set