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

Problem

(2*cos(2*x))/sin(2*x)=cot(x)−tan(x)

Solution

1. Identify the left side of the equation and apply the double angle identities for cosine and sine.

cos(2*x)=cos2(x)−sin2(x)

sin(2*x)=2*sin(x)*cos(x)

2. Substitute these identities into the left side of the expression.

(2*(cos2(x)−sin2(x)))/(2*sin(x)*cos(x))

3. Simplify the expression by canceling the common factor of 2 in the numerator and denominator.

(cos2(x)−sin2(x))/(sin(x)*cos(x))

4. Split the fraction into two separate terms by dividing each part of the numerator by the denominator.

cos2(x)/(sin(x)*cos(x))−sin2(x)/(sin(x)*cos(x))

5. Reduce each fraction by canceling common trigonometric factors.

cos(x)/sin(x)−sin(x)/cos(x)

6. Apply the quotient identities where cot(x)=cos(x)/sin(x) and tan(x)=sin(x)/cos(x)

cot(x)−tan(x)

Final Answer

(2*cos(2*x))/sin(2*x)=cot(x)−tan(x)

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