Question

The question is:
Set up a 2 column proof to show that each of the equations is an identity. Transform the left side to become the right side.
a. (tan + cot)^2 = sec^2 + csc^2

I'm having trouble with this.

b. (cos + sin)/cos + (cos - sin)/sin = csc sec

I'm having trouble with this too.

Answer 1

a)
you will need two identities here:
tan^2 x + 1 = sec^2 x , and
cot^2 x + 1 = csc^2 x

LS = tanx + cotx)^2
= tan^2x + 2tanxcotx + cot^2x
= sec^2x - 1 + 2 + csc^2x , because (tanx)(cotx) = 1
= sec^2 x + csc^2 x
= RS

b) LS = (sinx(cosx + sinx) + cosx(cosx - sinx))/(sinxcosx)
= (sinxcosx + sin^2x + cos^2x - sinxcosx)/(sinxcosx)
= 1/sinxcosx
= (1/sinx)(1/cosx)
= cscx secx
= RS