Question

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

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 .

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 .