Can someone help me solve this? Thanks!

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2 Answers
Sep 25, 2017

To verify an identity, one should make changes to only 1 side, until it is identical to the other side; I shall make changes to only the left.

Explanation:

Given:

sec(theta)/(csc(theta)-cot(theta))-sec(theta)/(csc(theta)+cot(theta)) = 2csc(theta)

Multiply the first fraction by 1 in the form (csc(theta)+cot(theta))/(csc(theta)+cot(theta))

sec(theta)/(csc(theta)-cot(theta))(csc(theta)+cot(theta))/(csc(theta)+cot(theta))-sec(theta)/(csc(theta)+cot(theta)) = 2csc(theta)

This makes the denominator become the difference of two squares:

(sec(theta)(csc(theta)+cot(theta)))/(csc^2(theta)-cot^2(theta))-sec(theta)/(csc(theta)+cot(theta)) = 2csc(theta)

This particular difference of two squares is the left side of the identity csc^2(theta)-cot^2(theta) = 1, therefore, the denominator becomes 1 and disappears:

sec(theta)(csc(theta)+cot(theta))-sec(theta)/(csc(theta)+cot(theta)) = 2csc(theta)

Use the identities sec(theta)csc(theta) =1 and sec(theta)cot(theta) = csc(theta):

1+csc(theta)-sec(theta)/(csc(theta)+cot(theta)) = 2csc(theta)

Multiply the next fraction by 1 in the form (csc(theta)-cot(theta))/(csc(theta)-cot(theta))

1+csc(theta)-sec(theta)/(csc(theta)+cot(theta))(csc(theta)-cot(theta))/(csc(theta)-cot(theta)) = 2csc(theta)

The produces the same difference of two squares, therefore, we shall merely delete the numerators:

1+csc(theta)-sec(theta)(csc(theta)-cot(theta)) = 2csc(theta)

Use the identities sec(theta)csc(theta) =1 and sec(theta)cot(theta) = csc(theta):

1+csc(theta)-1+csc(theta) = 2csc(theta)

Combine like terms:

2csc(theta) = 2csc(theta) Q.E.D.

Sep 27, 2017

Please see below

Explanation:

sectheta/(csctheta-cottheta)-sectheta/(csctheta+cottheta)

= (sectheta(csctheta+cottheta)-sectheta(csctheta-cottheta))/(csc^2theta-cot^2theta)

= sectheta(csctheta+cottheta-csctheta+cottheta) -> as csc^2theta-cot^2theta=1

= secthetaxx2cottheta

= 1/costhetaxx2costheta/sintheta

= 2/sintheta=2csctheta