Combine csc(2x)+cot(2x)

Problem

csc(2*x)+cot(2*x)

Solution

1. Rewrite the expression using the fundamental trigonometric identities for cosecant and cotangent.

csc(2*x)=1/sin(2*x)

cot(2*x)=cos(2*x)/sin(2*x)

2. Combine the fractions since they share a common denominator.

1/sin(2*x)+cos(2*x)/sin(2*x)=(1+cos(2*x))/sin(2*x)

3. Apply the double-angle identities for sine and cosine to simplify the numerator and denominator.

cos(2*x)=2*cos2(x)−1

sin(2*x)=2*sin(x)*cos(x)

4. Substitute these identities into the expression.

(1+(2*cos2(x)−1))/(2*sin(x)*cos(x))

5. Simplify the numerator by canceling the constants.

(2*cos2(x))/(2*sin(x)*cos(x))

6. Cancel the common factors of 2 and cos(x) from the numerator and denominator.

cos(x)/sin(x)

7. Identify the resulting expression as the cotangent function.

cos(x)/sin(x)=cot(x)

Final Answer

csc(2*x)+cot(2*x)=cot(x)

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