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

Problem

1+cot2(θ)=csc2(θ)

Solution

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

cot(θ)=cos(θ)/sin(θ)

csc(θ)=1/sin(θ)

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

1+(cos(θ)/sin(θ))2

3. Apply the power to the numerator and denominator.

1+cos2(θ)/sin2(θ)

4. Find a common denominator to combine the terms.

sin2(θ)/sin2(θ)+cos2(θ)/sin2(θ)

5. Combine the fractions over the common denominator.

(sin2(θ)+cos2(θ))/sin2(θ)

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

1/sin2(θ)

7. Recognize that the resulting expression is the definition of the cosecant squared.

csc2(θ)

Final Answer

1+cot2(θ)=csc2(θ)

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