Question

(cosec theta -sin theta ) ( sec theta - cos theta) ( tan theta + cot theta ) = 1

Answer 1

To prove,
(cosec θ - sin θ)(sec θ - cos θ)(tan θ + cot θ)=1

Proof:
LHS =(1/sin θ - sin θ)(1/cos θ - cos θ)(tan θ+1/tan θ)
         =(1-sin²θ)/sinθ (1-cos²θ)/cosθ(1+tan²θ)/tanθ
         =(cos²θ)/sinθ (sin²θ)/cosθ sec²θ/tanθ
         =cos θ sin θ (1/cos²θ)/(sinθ/cosθ)
         =cos θ sin θ(1/cos²θ)(cosθ/sinθ)
         =cos θ sin θ (cos θ/sin θ cos²θ)
         =cos θ sin θ (1/sin θ cos θ)
         =cos θ sin θ/sin θ cos θ
         =1=RHS
            ∴ Hence proved

                  Hope it Helps!!

Answer 2

Step-by-step explanation:

Given exp.

(

cosθ

1

−cosθ)(

sinθ

1

−sinθ)(

sinθ

cosθ

+

cosθ

sinθ

)

=

sin

2

θcos

2

θ

sin

2

θcos

2

θ

=1