Verify the Identity tan(pi/2-x)=cot(x)

Problem

tan(π/2−x)=cot(x)

Solution

1. Apply the ratio identity for tangent to rewrite the left side in terms of sine and cosine.

tan(π/2−x)=sin(π/2−x)/cos(π/2−x)

2. Use the cofunction identity for sine, which states that sin(π/2−θ)=cos(θ)

sin(π/2−x)=cos(x)

3. Use the cofunction identity for cosine, which states that cos(π/2−θ)=sin(θ)

cos(π/2−x)=sin(x)

4. Substitute these results back into the fraction.

cos(x)/sin(x)

5. Identify the resulting expression using the cotangent ratio identity.

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

Final Answer

tan(π/2−x)=cot(x)

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