How does one verify sinx-cscx=-cos^2x/sinx?

sinx-cscx=-cos^2x/sinx

3 Answers
Apr 3, 2018

See below.

Explanation:

It's best to leave one side alone, and simplify another side.

Let's leave the right alone, while simplifying the left.

Recall that cscx=1/x:

sinx-1/sinx=-cos^2x/sinx

Subtract the expressions on the left, taking sinx as the common denominator.

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

Now, recall the identity

sin^2x+cos^2x=1

This identity also tells us that

sin^2x-1=-cos^2x. So, we can use this to simplify our numerator on the left:

=cos^2x/sinx=-cos^2x/sinx

So, the identity holds true.

Apr 3, 2018

See below

Explanation:

We want to prove that sinx-cscx=-cos^2x"/"sinx. We use the following identity

cscx=1/sinx

So

sinx-cscx

=sinx-1/sinx

=(sin^2x-1)/sinx

=-(1-sin^2x)/sinx

=-cos^2x/sinx, as required. square

Apr 4, 2018

See method below

Explanation:

cscx=1/sinx (definition)

We want to show that sinx-1/sinx=-cos^2x/sinx

Since both terms contain sinx we can factorize

sinx-1/sinx = 1/sinx(sin^2x-1)

Substitute 1 using the identity 1=cos^2x + sin^2x

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

sin^2x cancels out.

1/sinx(-cos^2x)=-cos^2x/sinx

Therefore sinx-1/sinx=-cos^2x/sinx