Question

1+sin-cos /1+sin+cos=tan theta/2

Answer 1

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HEAR IS YOUR ANSWER:
Let’s use x as our variable instead of €so that we wish to prove

sinx−cosx+1sinx+cosx−1=1secx−tanx instead.

Now doesn’t the problem seem easier already?

Furthermore, let s=sinx,c=cosxfor simplicity.

Now sinx−cosx+1sinx+cosx−1=s−c+1s+c−1=(s−c+1)(1−s)c(s+c−1)c(1−s)=c1−s⋅1+sc−c−s2sc−c+c2=c1−s=1secx−tanx,

where the penultimate equality follows from the fact that 1−c2=s2.

The key step of the proof, namely multiplying by (1−s)cc(1−s), may seem like pulling a rabbit out of a hat, but the motivation behind it is simple.

The LHS of the identity is c1–s, so we try multiplying the RHS by c1−s⋅1−scand saving the factor c1–s.

I will reiterate my simple method for proving elementary contrived trig identities:

Convert everything into sines and cosinesSimplify using common trig identities

As long as you use this strategy, problems like these will become trivial.