Verify the Identity (sin(3x))/(sin(x)cos(x))=4cos(x)-sec(x)

Problem

sin(3*x)/(sin(x)*cos(x))=4*cos(x)−sec(x)

Solution

1. Apply the triple angle identity for the numerator sin(3*x) using the formula sin(3*x)=3*sin(x)−4*sin3(x)

(3*sin(x)−4*sin3(x))/(sin(x)*cos(x))

2. Factor out sin(x) from the numerator to prepare for simplification.

(sin(x)*(3−4*sin2(x)))/(sin(x)*cos(x))

3. Cancel the common factor sin(x) from the numerator and the denominator.

(3−4*sin2(x))/cos(x)

4. Use the Pythagorean identity sin2(x)=1−cos2(x) to rewrite the expression in terms of cos(x)

(3−4*(1−cos2(x)))/cos(x)

5. Distribute and simplify the terms in the numerator.

(3−4+4*cos2(x))/cos(x)

(4*cos2(x)−1)/cos(x)

6. Split the fraction into two separate terms.

(4*cos2(x))/cos(x)−1/cos(x)

7. Simplify each term using the definition of the secant function sec(x)=1/cos(x)

4*cos(x)−sec(x)

Final Answer

sin(3*x)/(sin(x)*cos(x))=4*cos(x)−sec(x)

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