Ask Me Help Desk - Prove sin2theta=2sinthetacostheta

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prove sin2theta=2sinthetacostheta

using the identities sin^2x+cos^2x=1, sin(x+y)=sinxcosy+cosxsiny, cos(x+y)=cosxcosy-sinxsiny prove the following 2 identities sin2theta=2sinthetacostheta and cos2theta=2cos^2theta-1

Identities:

$sin^2(x) + cos^2(x) = 1$

$sin (x + y) = sin(x)cos(y) + cos(x)sin(y)$

$cos (x + y) = cos(x)cos(y) - sin(x)sin(y)$

Prove:

$sin(2x) = 2sin(x)cos(x)$

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

The trick is to make x and y equal in the second two identities, and then to use the first identity to reduce.

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