Question

Prove that cot theta minus tan theta is equal to 2 cos squared theta minus one/ sin theta cos theta

Answer 1

Answer:

To prove

cotθ - tanθ = (2cos^θ -1)/sinθcosθ

Let LHS = cotθ - tanθ and RHS = (2cos^θ -1)/sinθcosθ

cotθ - tanθ = cosθ/sinθ  - sinθ /cosθ

          = [cosθ /cosθ x  cosθ/sinθ ] - [sinθ /sinθ  x sinθ /cosθ ]

          = cos^2 θ/ sinθcosθ - sin^2 θ /sinθcosθ

          = [cos^2 θ - sin^2 θ]/ sinθcosθ

          = [cos^2 θ - (1 - cos^2 θ)]/ sinθcosθ    (Since  cos^2 θ + sin^2 θ = 1)

          =[cos^2 θ - 1 + cos^2 θ]/ sinθcosθ

          = [2cos^2 θ - 1 ]/ sinθcosθ =  RHS

Therefore,

cotθ - tanθ = (2cos^θ -1)/sinθcosθ

Answer 2

Answer:

To Prove

cot Ф - tan Ф =  (2 cos²Ф -1) /(sinФcosФ)

Taking L.H.S to solve it

L.H.S=cot Ф - tan Ф

as we know that

cot Ф = cos Ф /  sin Ф

tan Ф = sin Ф / cos Ф

so the given becomes

L.H.S = (cos Ф / sin Ф) - (sin Ф / cos Ф)

Solving it gives

L.H.S= (cos² Ф - sin²Ф)/(sinФcosФ)

       as sin²Ф=1-cos²Ф

so

L.H.S=(cos²Ф-1+cos²Ф)/(sinФcosФ)

        =(2cos²Ф-1)/(sinФcosФ)

which is same as R.H.S

so this is the proof of the given