Simplify tan(x+pi/2)

Problem

tan(x+π/2)

Solution

1. Apply the sine and cosine identity to rewrite the tangent function in terms of sine and cosine.

tan(x+π/2)=sin(x+π/2)/cos(x+π/2)

2. Use the sum identities for sine and cosine, where 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)

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

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

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

4. Divide the results to find the simplified form of the tangent expression.

tan(x+π/2)=cos(x)/(−sin(x))

5. Apply the cotangent identity where cot(x)=cos(x)/sin(x)

tan(x+π/2)=−cot(x)

Final Answer

tan(x+π/2)=−cot(x)

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