How do you verify (tan^2x + 1) / tan^2x = csc^2x?

2 Answers
Jun 19, 2016

Use the Pythagorean identity tan^2x + 1 = sec^2x to start the simplification on the left side.

Explanation:

sec^2x/tan^2x =

Recall that sec^2x = 1/cos^2x, and that tan^2x = sin^2x/cos^2x

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

1/cos^2x xx cos^2x/sin^2x =

1/sin^2x =

Remember that 1/sinx = cscx

csc^2x =

Identity proved!!!

Hopefully this helps!

Jun 19, 2016

The definition of tangent is such that

tan(x)=sin(x)/cos(x), then the left side of the equation is

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

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

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

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

and, using the fundamental property of sin and cos that sin(x)^2+cos(x)^2=1 we have

(sin(x)^2+cos(x)^2)/(sin(x)^2) = 1/(sin(x)^2)=csc(x)^2.