please verify?

Question

Verify the equation:
cot^2(x)sec^2(x)-cot^2(x)=1

Answer 1

the simplest thing is to rewrite in terms of sine and cosine using the equations
cot(x) = cos(x)/sin(x)
sec(x) = 1/cos(x).

then you get

cos^2(x)/(sin^2(x)*cos^2(x)) - cos^2(x)/sin^2(x)

cancel the cos^2(x) in the first term and combine the two terms to get
(1 - cos^2(x))/sin^2(x)

and use the fact that 1-cos^2(x) = sin^2(x) to get
sin^2(x)/sin^2(x) = 1.

Answer 2

Change everything in terms of sin(x) and cos(x).

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

1/sin^2(x)- (cos^2(x)/sin^2(x)) = 1 Then multiply by sin^2(x)

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

This is one of the most basic trig identities, so
1=1

Answer 3

LHS
= (cos²x / sin²x).[ sec²x - 1 ]
= (cos²x / sin²x).[ 1 /cos²x - 1 ]
= (cos²x / sin²x.[ (1 - cos²x) / cos²x ]
= (cos²x / sin²x).[ sin²x / cos²x ]
= 1
RHS
= 1
LHS = RHS