Question

Given ONLY the fact that sin^2(θ) = (1-cos(2θ)) /2 and cos^2(θ) + sin^2(θ) = 1.

Show that: cos^2(θ)=1/2 + 1/2 cos(2θ)

Answer 1

Alright, so given what you have, Im going to take your 2nd equation of
\[\sin ^{2}\theta+\cos ^{2}\theta = 1\] and replace sin^2 with your first identity to give me:
\[\frac{ 1-\cos(2\theta) }{ 2 }+\cos ^{2}\theta=1\]I'll subtract both sides by our 1-cos(2x) portion to get:
\[\cos ^{2}\theta=1-\frac{ 1-\cos(2\theta) }{ 2 }\]I can combine the right side of the equation into one fraction by making 2 a common denominator:
\[\cos ^{2}\theta=\frac{ 2-(1-\cos(2\theta)) }{ 2 }\]I mae sure Im careful with signs here. All of the (1-cos2x) expression is being subtracted, so having those parenthesis is necessary to show this. Now I distribute the negative and combine the remaining constant terms:
\[\cos ^{2}\theta=\frac{ 1+\cos(2\theta) }{ 2 }\]Now all I do is separate this into two fractions, one for each term in the numerator:
\[\cos ^{2}\theta=\frac{ 1 }{ 2 }+\frac{ \cos(2\theta) }{ 2 }\]

Answer 2

OH! I see the mistake I've done. Thank you so much I really appreciate your help. :)

Answer 3

No problem ^_^