Question

Prove the following:
1 + cos x + sin x / 1+ cos x - sinx = 1+ sinx / cos x

Answer 1

LHS = ( 1 + cosx + sinx )/( 1 + cosx - sinx )

= (1 + cosx + sinx )(1 + cosx + sinx)/( 1 + cosx -sinx )(1 + cosx + sinx)

= ( 1 + cosx + sinx )²/{(1 + cosx) ² - (sinx)²}

= ( 1 + sin²x + cos²x + 2sinx.cosx + 2sinx + cosx )/{ 1 + cos²x +2cosx - sin²x }

[use , sin²x + cos²x = 1 ]

= ( 2 + 2sinx.cosx + 2sinx. + 2cosx )/2cosx(1 + cosx )

= 2( 1 + sinx )(1 + cosx )/2cosx(1 +cosx)

= ( 1 + sinx)/cosx = RHS

Answer 2

Hey there!

LHS = ( 1 + cosx + sinx )/( 1 + cosx - sinx ) 
= (1 + cosx + sinx )(1 + cosx + sinx)/( 1 + cosx -sinx )(1 + cosx + sinx)
= ( 1 + cosx + sinx )^2/{(1 + cosx)^2 - (sinx)^2} 
= ( 1 + sin^2x + cos^2x + 2sinx.cosx + 2sinx + cosx )/{ 1 + cos^2x +2cosx - sin^2x } 
= ( 2 + 2sinx.cosx + 2sinx. + 2cosx )/2cosx(1 + cosx )
= 2( 1 + sinx )(1 + cosx )/2cosx(1 +cosx)
= ( 1 + sinx)/cosx = RHS

Hope it helps You!