1+sin∅+cos∅÷1+sin∅-cos∅ = cos∅÷2

Question

Show that 1+sin∅+cos∅÷+sin∅-cos∅ = cos∅÷2

Answer 1

There is a typo in the problem. The right hand side should be cot(ø/2) not cosø/2. Then the identity works.

Using the half angle formula and x instead of ∅:

(1 + sin(x) + cos(x))/(1+sin(x) - cos(x))

= (1+2sin(x/2)cos(x/2) + cos²(x/2)-sin²(x/2))/(1+2sin(x/2)cos(x/2)-cos²(x/2)+sin²(x/2))

1-sin²(x/2) = cos²(x/2) and 1-cos²(x/2) = sin²(x/2)

This reduces the equation down to:

(2sin(x/2)cos(x/2) + 2cos²(x/2))/(2sin(x/2)cos(x/2) + 2sin²(x/2))

= ((cos(x/2)(sin(x/2) + cos(x/2))/(sin(x/2)(sin(x/2)+cos(x/2))

= cos(x/2)/sin(x/2) = cot(x/2)

Answer 2

∅=30°

1+sin30+cos30+sin30-cos30=(cos30)/2

1+2sin30=(cos30)/2

1+1 =.43301.........??????????????????