Verify the following identity. Show all your work. Don't skip any steps.

Question

Do not change the right side of the equation, only the left side

csc^2 theta * tan^2 theta-1= tan^2 theta

Answer 1

Starting with the left side of the equation:

csc^2(theta) * tan^2(theta) - 1

= (1/sin^2(theta)) * (sin^2(theta)/cos^2(theta)) - 1 [using the trig identity tan(theta) = sin(theta)/cos(theta) and csc(theta) = 1/sin(theta)]

= 1/cos^2(theta) - 1 [the sin^2(theta) terms cancel out]

= (1-cos^2(theta))/cos^2(theta)

= sin^2(theta)/cos^2(theta)

= tan^2(theta)

Therefore, the left side of the equation is equal to the right side, so the identity is verified.