Question 396065
instruction: show that  the following will give the value of 1.

prob. 1.  (csc x - cot x)(csc x + cot x)

 prob. 2. 1 divided by 1 + tan squared x + 1 divided by cot squared x + 1

prob. 3.  1 - sin squared x divided by cos squared x.


sohcahtoa:
sin = opp/hyp, cos = adj/hyp, tan = opp/adj,
tan = sin/cos, cot = 1/tan = cos/sin,
csc = 1/sin, sec = 1/cos,
sin^2 + cos^2 = 1, 1 + tan^2 = sec^2, 1 + cot^2 = csc^2


prob. 1.  (csc x - cot x)(csc x + cot x)
(csc x - cot x)(csc x + cot x)
(1/(sin x) - (cos x)/(sin x))(1/(sin x) + (cos x)/(sin x))
by FOIL (First Outer Inner Last) middle terms are gonna cancel out
1/(sin^2 x) - (cos^2 x)/(sin^2 x)
(1 - cos^2 x)/(sin^2 x)
(sin^2 x)/(sin^2 x)
1
prob. 2. 1 divided by 1 + tan squared x + 1 divided by cot squared x + 1
1/(1 + tan^2 x) + 1/(cot^2 x + 1)
1/(sec^2 x) + 1/(csc^2 x)
cos^2 x + sin^2 x
1
prob. 3.  1 - sin squared x divided by cos squared x.
(1 - sin^2 x)/(cos^2 x)
(cos^2 x)/(cos^2 x)
1