Question

Prove this identity

1. csc^2(pi/2-x)=1+sin^2xcsc^2(pi/2-x)
2. tan(pi/4+x)+tan(pi/4-x)=2sec2x

Answer 1

I will do the first one.

LS = 1/ sin^2(pi/2 - x)
= 1/(sin(pi/2)cosx = cos(pi/2)sinx)^2
= 1/(cosx)^2

RS = 1 + sin^2x/sin^2(pi/2 - x)
= 1 + sin^2x/cos^2x --- we did that part in LS
= (cos^2 + sin^2x)/cos^2x --- common denominator etc
= 1/cos^2x
= LS

for the second I would change the tan to sin/cos and use the expansion formulas

give it a try.