Simplify the trigonometric expression and identify the trigonometric identities used: cos^2(θ)-1 / cos^2(θ) - cosθ

Question

cos²(θ)-1 / cos^2(θ) - cosθ

Answer 1

I read the question a little differently from Minoo. I saw the denominator as :

Cos^2 x - cos x , not cos2x-cosx

In that case, the numerator factors to (cos x- 1)(cos x +1)

Ands the denominator factors to cos x (cos x- 1)

This simplifies to (cos x + 1)/cos x or 1 + sec x

I used x instead of theta for convenience

Answer 2

Hi,

cos²(θ)-1 / cos^2(θ) - cosθ

First of all you need to know : #1 a²- b²= (a- b)(a + b) #2 cos2θ = 2cos²θ - 1

Now: cos²(θ)-1 = ( cosθ -1 ) ( cosθ +1) base of #1 AND denominator cos^2(θ) - cosθ = 2cos²θ - 1 - cosθ

replace to the given fraction: cos²(θ)-1 / [ cos^2(θ) - cosθ ]= ( cosθ -1 ) ( cosθ +1) / [2cos²θ - 1 - cosθ]

you can change denominator to the ( cosθ -1 ) ( 2cosθ +1) then simplify given fraction:

cos²(θ)-1 / [cos^2(θ) - cosθ ]= ( cosθ -1 ) ( cosθ +1)/ ( cosθ -1 ) ( 2cosθ +1) you can simplify ( cosθ -1 ) of top and bottom then you will have ( cosθ +1)/ ( 2cosθ +1)

I hope it was useful,

Minoo