Question

1+sin x-cos x/1+sin x +cos x +1+sin x +cos x /1+sin x - cos x =2cosec x

Answer 1

(1 + sin - cos) (1 + sin + cos)

----------------- + ------------------ = 2cosec get common denominator

(1 + sin + cos) (1 + sin - cos)

(1 + sin - cos)2 + (1 + sin + cos)2

---------------------------------------- = 2cosec expand all expressions

(1 + sin + cos)(1 + sin - cos)

(1 + sin - cos + sin + sin2 - sincos - cos - sincos + cos2) +

(1 + sin + cos + sin + sin2 + sincos + cos + sincos + cos2)

---------------------------------------------------------------------------------------- = 2cosec

(1 + sin - cos + sin + sin2 - sincos + cos + sincos - cos2)

Now we're going to combine like terms, remembering that sin2 + cos2 = 1 and 1 - cos2 = sin2

(2 + 2sin - 2cos - 2sincos) + (2 + 2sin + 2cos + 2sincos)

------------------------------------------------------------------ = 2cosec

(2sin2 + 2sin)

(4 + 4sin)

--------------- = 2cosec

2sin(sin + 1)

4(sin + 1)

--------------- = 2cosec

2sin(sin + 1)

2/sin = 2cosec

2cosec = 2cosec

I hope it wasn't too hard to follow. Good luck.