1/cos x + 1/sin x

Question

The directions are to solve. Not sure if I can leave the sin and cis in the denominator.   Unit on trig identities

Answer 1

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

2(sin(x)+cos(x))/sin(2x). “≡” means “identical to”.

Now sin(x)+cos(x)≡(cos²(x)-sin²(x))/(cos(x)-sin(x))≡

cos(2x)/(cos(x)-sin(x)).

But cos(2x)/sin(2x)≡cot(2x).

So 2(sin(x)+cos(x))/sin(2x)≡

2cot(2x)/(cos(x)-sin(x)).

So 1/sin(x)+1/cos(x)≡2cot(2x)/(cos(x)-sin(x)).

This is one trig identity but there may be others.