Question

prove that :- cos theta by 1-sin theta = 1 + sin theta by cos theta​

Answer 1

Answer:

» cosØ / (1 - sinØ) = (1 + sinØ) / cosØ

» cos²Ø = 1 - sin²Ø

» sin²Ø + cos²Ø = 1

It's an identity.

Therefore given expression is correct.

Answer 2

Answer:

Consider the LHS.

⇒

sinθ+cosθ−1

sinθ−cosθ+1

Divide numerator and denominator with cosθ.

⇒

cosθ

sinθ

+

cosθ

cosθ

−

cosθ

1

cosθ

sinθ

−

cosθ

cosθ

+

cosθ

1

⇒

tanθ+1−secθ

tanθ−1+secθ

⇒

tanθ−secθ+1

tanθ+secθ−1

Put sec

2

θ−tan

2

θ=1 in the numerator.

⇒

tanθ−secθ+1

(tanθ+secθ)−(sec

2

θ−tan

2

θ)

⇒

tanθ−secθ+1

(secθ+tanθ)[1−secθ+tanθ]

⇒secθ+tanθ

Multiply and divide the above result with (secθ−tanθ).

⇒secθ+tanθ

⇒

cosθ

1

+

cosθ

sinθ

⇒

cosθ

1+sinθ

Hence, LHS=RHS.