prove 1/sinx-sinx=cos^2xcscx

please help its due tomorrow

work on the left side

LS = 1/sinx - sinx
= (1 - sin^2 x)/sinx , after finding a common denominator of sinx
= (cos^2 x)/sinx , because sin^2 x + cos^2 x = 1
= (cos^2 x)(1/sinx)
= (cos^2 x)(cscx)
= RS

To prove the equation 1/sin(x) - sin(x) = cos^2(x)csc(x), we will manipulate the left-hand side and right-hand side separately until they are equivalent.

Starting with the left-hand side (LHS):

1/sin(x) - sin(x)

First, we need to find a common denominator for these terms. The common denominator for 1/sin(x) and sin(x) is sin(x). Therefore, we can re-write the equation as:

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

Now we can simplify the numerator by using the identity sin^2(x) = 1 - cos^2(x):

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

Simplifying further:

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

(cos^2(x))/sin(x)

Now let's simplify the right-hand side (RHS):

cos^2(x)csc(x)

Recall that csc(x) is the reciprocal of sin(x), so csc(x) = 1/sin(x). We can substitute this into the equation:

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

Using the rule of multiplication, we can rewrite this as:

cos^2(x)/sin(x)

We can see that the right-hand side (RHS) is equal to the simplified form of the left-hand side (LHS). Therefore, we can conclude that the equation is proved:

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