Prove 1 + cos (2 θ) 2 / cos (θ) = cos (θ) is an identity

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Mathematics

Elisha Huang

23 December, 15:54

Prove 1 + cos (2 θ) 2 / cos (θ) = cos (θ) is an identity

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Mike Summers · Answer 1

prove [1 + cos (2 θ) ] / 2cos (θ) = cos (θ) is an identity

[ 1 + cos (2 θ) ] / 2cos (θ)

cos (2θ) = 2 (cos θ) ^2 - 1

so plug 2 (cos θ) ^2 - 1 in for cos (2 theta)

[ 1 + 2 (cos θ) ^2 - 1 ] / 2cos (θ)

= [ 2 (cos θ) ^2 ] / 2cos (θ)

= [ (cos θ) ^2 ] / cos (θ)

= (cos θ)

so (cos θ) = (cos θ) is true. Proven.

Step-by-step explanation:

prove [1 + cos (2 θ) ] / 2cos (θ) = cos (θ) is an identity

...

we need to show that the expression on the left side will eventually transform into cos (Θ)

Start with [1 + cos (2 θ) ] / 2 cos (θ)

prove [1 + cos (2 θ) ] / 2cos (θ) = cos (θ) is an identity

[ 1 + cos (2 θ) ] / 2cos (θ)

cos (2θ) = 2 (cos θ) ^2 - 1

so plug 2 (cos θ) ^2 - 1 in for cos (2 theta)

[ 1 + 2 (cos θ) ^2 - 1 ] / 2cos (θ)

= [ 2 (cos θ) ^2 ] / 2cos (θ)

= [ (cos θ) ^2 ] / cos (θ)

= (cos θ)

so (cos θ) = (cos θ) is true. Proven.