show that (1+sin?)/cos? = cos?/(1-sin?)

Question

(1+sin theta)/cos theta is equal to cos theta/(1-sin theta) pls i hav tried buh it seems undefined

Answer 1

Working the left side of the equation:

 

Multiply top and bottom by the conjugate of the top {1 - sin(θ)}.

 

that gives you:

= 1 - sin2(θ) / cos(θ)[1 - sin(θ)]

= cos2(θ) / cos(θ)[1 - sin(θ)]

= cos(θ) / 1 - sin(θ)

 

Conversely working the right side of the equation:

Multiply top and bottom by 1 + sin(θ) ...

Answer 2

I would recommend multiplying the equation by the product of the two denominators, cosθ(1-sinθ), which will immediately get the fractions out of the picture.

 

(1+sinθ)/cosθ = cosθ/(1-sinθ)

 

cosθ(1-sinθ)(1+sinθ)/cosθ = cosθ(1-sinθ)cosθ/(1-sinθ)

 

which allows us to factor out common terms on each side

 

(1-sinθ)(1+sinθ) = cosθ cosθ

 

which is easily rewritten

1 - sin²θ = cos²θ

 

Isolate the 1 by adding sin²θ to both sides

 

1 = sin²θ + cos²θ

 

which is is the fundamental trigonometric identity.  Since we know this to be true, we know the original equation to be true.