Question

Assignment
1. Prove :
sin theta plus cosec theta upon sine theta is equal to 2 + cot.​

Answer 1

Answer:

We have to prove that

cot+cosecθ

sinθ

=2+

cotθ−cosecθ

sinθ

or,

cotθ+cosecθ

sinθ

−

cotθ−cosecθ

sinθ

=2

Now,

⇒ LHS =

cotθ+cosecθ

sinθ

−

cotθ−cosecθ

sinθ

⇒ LHS =

cosecθ+cotθ

sinθ

+

cosecθ−cotθ

sinθ

⇒ LHS = sinθ{

cosecθ+cotθ

1

+

cosecθ−cotθ

1

}

⇒ LHS = $$sin \, \theta\left\{\dfrac{cosec \, \theta \, - \, cot \, \theta \, + \, cosec \, \theta \, + \, cot \, \theta}{cosec^2 \, \theta \, - \, cot^ \, \theta}\right\} \, = \, sin \, \theta \, \left(\dfrac{2 \, cosec \, \theta}{1} \right)$$

⇒ LHS = sinθ(2cosecθ)=2sinθ×

sinθ

1

=2=RHS

⇒ LHS = 2 = RHS

ALTERNANATIVELY,

LHS =

cotθ+cosecθ

sinθ

⇒LHS=sinθ(cosecθ−cotθ) [∵

cosecθ+cotθ

1

=cosecθ−cotθ]

⇒ LHS = sinθ(

sinθ

1

−

sinθ

cosθ

)=sinθ(

sinθ

1−cosθ

)

⇒ LHS = 1 - cos θ

⇒ = 2 - (1 + cos θ)

⇒ LHS = 2 -

1−cosθ

(1+cosθ)(1−cosθ)

⇒ LHS = 2 -

1−cosθ

(1−cos

2

θ)

⇒ LHS = 2 -

1−cosθ

sin

2

θ

=2−

sinθ

1−cosθ

sinθ

=2−

sinθ

1

−

sinθ

cosθ

sinθ

⇒ LHS = 2 -

cosecθ−cotθ

sinθ

= RHS

Answer 2

Answer:

refers to the Attachment

hope it helps u

#NAWABZAADI