Question

Prove that cot theta -tan theta = 2cos^2 theta-1/sintheta× costheta

Answer 1

Answer: Proved

Step-by-step explanation:

Cot Ф - Tan Ф = 2 cos ²Ф - 1/ sinФcosФ

L.H.S = Cos Ф/sinФ - sinФ/cosФ

= cos²Ф - sin Ф/sinФ. cosФ

=cos²Ф - (1- cos Ф)/sin Ф.cosФ

= cos² Ф - 1 + cos²Ф/sinФcosФ

=2 cosФ -1 / sinФ cosФ

Answer 2

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