Simplify cot(x)^2+1

Problem

cot(x)+1

Solution

1. Identify the trigonometric identity for the cotangent function in terms of sine and cosine.

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

2. Substitute the identity into the expression and square the terms.

cot(x)+1=cos(x)/sin(x)+1

3. Find a common denominator to combine the fraction and the integer.

cos(x)/sin(x)+sin(x)/sin(x)

4. Combine the numerators over the common denominator.

(cos(x)+sin(x))/sin(x)

5. Apply the Pythagorean identity which states that sin(x)+cos(x)=1

1/sin(x)

6. Recognize the reciprocal identity for the cosecant function.

1/sin(x)=csc(x)

Final Answer

cot(x)+1=csc(x)

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