Prove that (cot a+cosec a)^2=(1+cos a)/(1-cos a)

Well, [cot(x)+csc(x)]^2=cot(x)^2 + 2cot(x)csc(x) +csc(x)^2
now, let's sub in sin and cos in place of cot and csc.

cos^2/sin^2 +2(cos/sin)(1/sin)+1/sin^2

now, we add it up because denominators are all sin^2.

(cos^2+2cos+1)/sin^2 = (cos(x)+1)^2/(1-cos^2)

note that I substituted the trig identity sin^2+cos^2=1 This in turn simplifies to:

(cos(x)+1)^2/[(1-cos)(1+cos)]

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

It's a neat little trick.