cosxcscx-2cosx=cscx-2

To solve this trigonometric equation, we'll start by simplifying both sides and bringing all terms to one side:

cos(x)csc(x) - 2cos(x) = csc(x) - 2

First, let's simplify cos(x)csc(x). Recall that the reciprocal of sine is cosecant, and the reciprocal of cosine is secant. Therefore, we have:

cos(x) * csc(x) = cos(x) * 1/sin(x) = cos(x)/sin(x) = cot(x)

Now, let's substitute this back into the equation:

cot(x) - 2cos(x) = csc(x) - 2

Next, we need to simplify the right side of the equation. Recall that the reciprocal of sine is cosecant:

csc(x) = 1/sin(x)

Substituting this back into the equation, we get:

cot(x) - 2cos(x) = 1/sin(x) - 2

Now, let's simplify the left side of the equation. Recall that cotangent is the reciprocal of tangent:

cot(x) = 1/tan(x)

Substituting this back into the equation, we get:

1/tan(x) - 2cos(x) = 1/sin(x) - 2

Next, let's simplify the right side of the equation. Recall that tangent is the ratio of sine to cosine:

tan(x) = sin(x) / cos(x)

Substituting this back into the equation, we get:

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

Multiplying the numerator and denominator of the left side by cos(x) to rationalize the denominator:

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

Simplifying the left side, we get:

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

Combining like terms, we have:

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

Now, let's convert the right side with a common denominator:

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

Simplifying, we get:

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

To get rid of the denominator, we can multiply both sides of the equation by sin(x):

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

Moving all terms to one side of the equation:

0 = 1 - 2sin(x) + sin(x)cos(x)

Finally, we have the equation:

0 = 1 - 2sin(x) + sin(x)cos(x)

At this point, the equation does not seem easily solvable analytically. To find an approximate solution, we can use numerical methods like graphing or iterative approaches.