How to prove cos^2x=(1+cos2x/2)?

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Thx

• cos 2x = cos(x+x) = cos x cos x- sin x sin x = cos^2 x – sin^2 x

(1 + cos 2x)/2 = (1 + cos^2 x – sin^2 x)/2 = (1 + cos^2 x – 1 + cos^2 x)/2 = 2 cos^2 x / 2 = cos^2 x

: )

• If you want you can start from : sin^2 x + cos^2 x = 1 Divide through by cos^2 x to get: (sin^2 x)/(cos^2 x) + 1 = 1/cos^2 x = tan^2x + 1 = sec^2x The power reduction formula: cos^2 x = (1 + cos 2x)/2, 2cos^2x = 1 + cos 2x Now, your problem: 1 + cos 2x = 2/(1 + tan^2 x) Using the fact that: sec^2x = 1 + tan^2x 1 + cos 2x = 2/(sec^2 x) 2 cos^2 x = 2/(1/cos^2 x) 2cos^2 x = 2cos^2 x Which proves the identity

RE:

How to prove cos^2x=(1+cos2x/2)?

Thx

• cos 2x

= cos (x + x)

= cosx cosx — sinxsinx

= cos^2x — sin^2x

= cos^2x — (1 — cos^2x)

= 2 cos^2 x — 1

OR 2 cos^2 x = 1 + cos 2x

OR cos^2x = (1 + cos 2x) / 2

• cos^2(x) – sin^2(x) = cos(2x)

cos^2(x) – (1 – cos^2(x) ) = cos(2x)

2cos^2(x) = cos(2x) + 1 => cos^2(x) = (cos(2x) + 1)/2

• cos^2(x) + sin^2(x)=1 (identity) implies: sin^2(x)= 1-cos^2(x)…..(1)

cos2x= cos^2(x)-sin^2(x)…….(2)

Replace (1) in (2) implies cos(2x)= cos^2(x) – [1-cos^2(x)]

cos(2x)=2cos^2(x)-1