Simplify tan(x-pi/2)

Problem

tan(x−π/2)

Solution

1. Identify the tangent subtraction identity, which states tan(A−B)=(tan(A)−tan(B))/(1+tan(A)*tan(B)) However, since tan(π/2) is undefined, use the sine and cosine definitions instead.

2. Rewrite the expression using the ratio of sine to cosine.

tan(x−π/2)=sin(x−π/2)/cos(x−π/2)

3. Apply the angle subtraction formulas sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B) and cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B)

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

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

4. Substitute the known values sin(π/2)=1 and cos(π/2)=0

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

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

5. Simplify the resulting fraction.

(−cos(x))/sin(x)=−cot(x)

Final Answer

tan(x−π/2)=−cot(x)

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