Question

Verify the identity

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Answer 1

see below

Explanation:

sin x 1

------------------- = -----------

sec^2 x - tan ^2 x csc x

Sec = 1/cos and tan = sin/cos

sin x 1

------------------- = -----------

1/ cos ^2 x -sin^2/cos ^2 x csc x

Factor the denominator

sin x 1

------------------- = -----------

(1-sin^2 x)/ cos ^2 x csc x

We know that 1 - sin^2 x = cos ^2

sin x 1

------------------- = -----------

(cos^2 x)/ cos ^2 x csc x

sin x 1

------------------- = -----------

1 csc x

Multiply the top and bottom of the left hand side by 1/ sin x

sin x * 1/ sin x 1

------------------- = -----------

1 * 1 sin x csc x

1 1

------------------- = -----------

1 sin x csc x

We know that 1/sin x = csc

1 1

--------- = -----------

csc (x) csc x