Prove that sinx * csc(-x) = -1

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hala718 eNotes educator | Certified Educator

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sinx * csc(-x) = -1

We will use the trigonometric identities to prove the identity.

First, we know that csc(x) = 1/sinx.

==> csc(-x) = 1/sin(-x)

Now we will substitute in the identity.

==> sinx * 1/sin(-x) = -1

Now, we know that sin(-x) = -sinx

==> sinx * 1/-sin(x) = -1

Now we will reduce similar terms.

==>  -1 = -1  .............q.e.d

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justaguide eNotes educator | Certified Educator

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We have cosec x = 1/sin x.

sin (-x) = -sin x.

Now sin x * cosec(-x)

=> sin x * (1 / sin (-x))

=> sin x * ( -1 / sin x)

=> -1 * (sin x / sin x)

= -1

Therefore we prove that sin x * cosec(-x) = -1

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