Verify the Identity 1+cot(x)^2=csc(x)^2

Problem

1+cot2(x)=csc2(x)

Solution

1. Recall the definitions of the trigonometric functions in terms of sine and cosine.

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

2. Substitute the definition of cotangent into the left side of the identity.

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

3. Apply the power to the numerator and denominator.

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 fractions over the common denominator.

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

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

1/sin2(x)

7. Recognize the definition of the cosecant function, where csc(x)=1/sin(x)

csc2(x)

Final Answer

1+cot2(x)=csc2(x)

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