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

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 .  

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