Question

Prove the identity.

Answer 1

\[\cos4\theta - 4\sin^2 \theta = 2(\cos2\theta + \frac{ 1 }{ 2 })^2 + \frac{ 7 }{ 2 }\]

Answer 2

https://www.wolframalpha.com/input/?i=%5Ccos(4%5Ctheta)+-+4%5Csin%5E2+%5Ctheta+%3D+2(%5Ccos(2%5Ctheta)+%2B+%5Cfrac%7B+1+%7D%7B+2+%7D)%5E2+%2B+%5Cfrac%7B+7+%7D%7B+2+%7D

Answer 3

\[\cos4 \theta - 4 \sin^2 \theta = 2(\cos2 \theta + \frac{ 1 }{ 2 })^2 - \frac{ 7 }{ 2 }\]

Answer 4

sorry it should be minus 7/2 not plus 7/2, my bad

Answer 5

@pythagoras123 Do you know the expansion of Cos(2theta)?

Answer 6

try using cos 4(theta) = 2 cos^2 (2theta) - 1

and sin^2 theta = 1 - cos^2 theta

Answer 7

so LHS = 2 cos^2 (2theta) - 1 - 4(1 - cos^2 theta)

= 2cos^2 (2theta) + 4 cos^2 (theta) - 5

Answer 8

Now expand RHS to see if it simplifies to the above.

Answer 9

2(cos 2theta + 0.5)^2 - 3.5
= 2( cos^2 (2theta) + cos(2theta) + 0.25) - 3.5
= 2cos^2 (2theta) + 2 cos(2theta) + 0.5 - 3.5
= 2 cos ^(2theta) + 2 cos (2 theta) - 3

Answer 10

now use the identity 2 cos(2 theta) = 4 cos^2 theta - 2
in the above and see what you get

Answer 11

I think you'll find it gives the same as the LHS

Answer 12

I guess you know what LHS and RHS are

- left hand side and right HS