Verify the Identity (tan(x)^2cos(x))/(2sec(x))=1/2*sin(x)^2

Problem

(tan2(x)*cos(x))/(2*sec(x))=1/2*sin2(x)

Solution

1. Rewrite the trigonometric functions in the numerator and denominator in terms of sine and cosine using the identities tan(x)=sin(x)/cos(x) and sec(x)=1/cos(x)

((sin(x)/cos(x))2*cos(x))/(2*(1/cos(x)))

2. Simplify the numerator by squaring the fraction and multiplying by cos(x)

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

sin2(x)/cos(x)/2/cos(x)

3. Divide the fractions by multiplying the numerator by the reciprocal of the denominator.

sin2(x)/cos(x)⋅cos(x)/2

4. Cancel the common factor of cos(x) from the numerator and the denominator.

sin2(x)/2

5. Factor out the constant to match the form of the right-hand side of the identity.

1/2*sin2(x)

Final Answer

(tan2(x)*cos(x))/(2*sec(x))=1/2*sin2(x)

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