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

Problem

cot(π/2−x)=tan(x)

Solution

1. Identify the cofunction identity for the cotangent function, which relates the trigonometric function of a complementary angle to its corresponding cofunction.

2. Apply the definition of the cotangent function in terms of sine and cosine to rewrite the left side of the equation.

cot(π/2−x)=cos(π/2−x)/sin(π/2−x)

3. Use the sum and 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(π/2−x)=cos(π/2)*cos(x)+sin(π/2)*sin(x)

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

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

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

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

5. Simplify the ratio by substituting these results back into the cotangent definition.

sin(x)/cos(x)=tan(x)

Final Answer

cot(π/2−x)=tan(x)

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