sinx(ccscx-sinx=cos^x

sin*csc=1

sin*(sin= sin^2x

1-sin^2x=cos^2 x

cos^2x=cos^2x

To solve the equation sin(x)(csc(x)-sin(x)) = cos^2(x), we can use some trigonometric identities and algebraic manipulation. Follow these steps:

Step 1: Start by simplifying the left side of the equation using the reciprocal identities. Recall that csc(x) is the reciprocal of sin(x) and sec(x) is the reciprocal of cos(x).

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

Step 2: Continue simplifying the left side by combining the terms inside the parentheses.

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

Since sin(x)/sin(x) simplifies to 1, the equation becomes:

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

Step 3: Apply the Pythagorean identity sin^2(x) + cos^2(x) = 1.

1 - sin^2(x) = cos^2(x) has transformed into:

cos^2(x) = cos^2(x)

Step 4: Both sides of the equation are equal, so the equation is satisfied for all values of x. This means that the original equation is true for all values of x.

Therefore, the solution to the equation sin(x)(csc(x)-sin(x)) = cos^2(x) is the set of all real numbers.