Simplify 1+cot(x)^2

Problem

1+cot2(x)

Solution

1. Identify the trigonometric identity involved. The expression 1+cot2(x) is one of the fundamental Pythagorean identities.

2. Recall the definition of the cotangent function in terms of sine and cosine:

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

3. Substitute the definition into the expression:

1+cos2(x)/sin2(x)

4. Find a common denominator to combine the terms:

sin2(x)/sin2(x)+cos2(x)/sin2(x)

5. Combine the numerators over the common denominator:

(sin2(x)+cos2(x))/sin2(x)

6. Apply the basic Pythagorean identity sin2(x)+cos2(x)=1

1/sin2(x)

7. Recognize the reciprocal identity for the cosecant function, where csc(x)=1/sin(x)

csc2(x)

Final Answer

1+cot2(x)=csc2(x)

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