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

Problem

cot(x−π/2)=−tan(x)

Solution

1. Apply the definition of the cotangent function in terms of sine and cosine.

cot(x−π/2)=cos(x−π/2)/sin(x−π/2)

2. Use the difference formulas for cosine and sine, where cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B) and sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

cos(x−π/2)=cos(x)*cos(π/2)+sin(x)*sin(π/2)

sin(x−π/2)=sin(x)*cos(π/2)−cos(x)*sin(π/2)

3. Substitute the known values for cos(π/2)=0 and sin(π/2)=1 into the expressions.

cos(x−π/2)=cos(x)*(0)+sin(x)*(1)=sin(x)

sin(x−π/2)=sin(x)*(0)−cos(x)*(1)=−cos(x)

4. Simplify the fraction by substituting these results back into the cotangent expression.

cot(x−π/2)=sin(x)/(−cos(x))

5. Identify the tangent identity where tan(x)=sin(x)/cos(x)

sin(x)/(−cos(x))=−tan(x)

Final Answer

cot(x−π/2)=−tan(x)

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