Simplify cot(x-pi/2)

Problem

cot(x−π/2)

Solution

1. Identify the trigonometric identity for the cotangent function in terms of sine and cosine.

cot(θ)=cos(θ)/sin(θ)

2. Substitute the given argument into the identity.

cot(x−π/2)=cos(x−π/2)/sin(x−π/2)

3. Apply the difference identities 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)

4. Evaluate the trigonometric values at π/2 noting that cos(π/2)=0 and sin(π/2)=1

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

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

5. Simplify the fraction by substituting the evaluated expressions back into the cotangent formula.

cot(x−π/2)=sin(x)/(−cos(x))

6. Recognize the definition of the tangent function, tan(x)=sin(x)/cos(x)

cot(x−π/2)=−tan(x)

Final Answer

cot(x−π/2)=−tan(x)

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