Understanding how to solve a trig , not being able to recreate past solution ! :(

[woven horizon]

This is the general way to solve the question

But I created my own maniupulation

I took cosA/1 + sinA + 1+sinA/cosA

see the second term , I can easily break it into two seperate terms

(1+sinA)/cosA --> 1/cosA + SinA/CosA --> secA + Tan A

easy .. I got one secA and TanA ... that means from the first term I gotta extract the term -tanA + secA

and for that I manipulated the term by adding conjugates so that I can make the lower term "single"

the entire idea of this manipulation I discovered revolved around the idea that we make the denominator single and then the terms just cancel out themselves like butter and we end up getting our answer in the way we want 😄 ... it works 99% of the time

I know 1 - sin^2 X = cos^2X

so my next step is multiplying cosA/(1 + SinA) by (1-SinA)/(1-SinA) 🙂

this gives me cosA + CosASinA / cos^2A

and that leaves me with secA - TanA

here is the working

So so so much simpler than the original method as given by the book solutions to solve it out 🙂 ... Thats what the idea of conjugate identity revolves around

now there is this question which Idk why has the internet overwhelmingly giving just one answer!

The real question -

/$ How can I prove that cosA-sinA+1/cosA+sinA-1=cosecA+cotA? $/

man I forgot that box thingy notation nevermind

but see the solution

maximum solutions have the answer as

divide the first step by SinA ... I get it , why we do it , whats the motivation to do it , we want to obtain cosec and cot right , so dividing by sin is a great manipulation trick but I used my conjugate method and it worked perffectly!

The difficulty tho was identifying the conjugates when I first solved it but I used another trick to deal with three terms

here is the answer