Question

Using the identity sec²θ= 1+ tan² θ, prove that

tanθ+secθ-1 1+sinθ
------------------- = ---------
tanθ-secθ +1 cosθ​

Answer 1

Step-by-step explanation:

Consider the LHS.

⇒

tanθ−secθ+1

tanθ+secθ−1

⇒

tanθ−secθ+1

tanθ+secθ−(sec

2

θ−tan

2

θ)

(∵sec

2

θ−tan

2

θ=1)

⇒

tanθ−secθ+1

(tanθ+secθ)−(secθ+tanθ)(secθ−tanθ)

(∵a

2

−b

2

=(a+b)(a−b))

⇒

tanθ−secθ+1

(tanθ+secθ)[1−(secθ−tanθ)]

⇒

tanθ−secθ+1

(tanθ+secθ)(tanθ−secθ+1)

⇒tanθ+secθ=

cosθ

sinθ

+

cosθ

1

=

cosθ

1+sinθ

Here,

LHS=RHS

Hence proved....