Jika tan α + tan β/sin(α + β) = 1/2, maka sec α . sec β adalah... A. -1/2 B. -1/4 C. 1/4 D. 1/3 E. 1/2

(Tan a + Tan b)/(sin (a + b)) = 1/2
2(Tan a + Tan b) = sin (a + b)
2((sin a)/(cos a) + (sin b)/(cos b)) = sin (a + b)
2(sin a cos b + cos a sin b)/(cos a sin a) = sin (a + b)
2 (sin a cos b + cos a sin b) = (cos a cos b) . sin (a + b)
2 sin (a + b) = (cos a cos b) . sin (a + b)
2 = cos a cos b

sec a . sec b = 1/(cos a) . 1/(cos b) = 1/(cos a cos b) = 1/2

[tex]$\begin{align} \frac{tan \ \alpha + tan \ \beta }{sin( \alpha + \beta )} &= \frac{1}{2} \\ \frac{\frac{sin \ \alpha }{cos \ \alpha } + \frac{sin \ \beta }{cos \ \beta } }{sin( \alpha + \beta )} &= \frac{1}{2} \\ \frac{\frac{sin \ \alpha \ .\ cos \ \beta \ + \ sin \ \beta \ . \ cos \ \alpha }{cos \ \alpha \ . \ cos \ \beta }}{sin( \alpha + \beta) } &= \frac{1}{2} \\ \frac{sin( \alpha + \beta) }{cos \ \alpha \ . \ cos \ \beta \ . \ (sin(\alpha + \beta))} &= \frac{1}{2} \\ \frac{1}{cos \ \alpha \ . \cos \ \beta } &= \frac{1}{2} \\ sec \ \alpha \ . \ cos \ \beta &= \frac{1}{2} \\\\ \boxed{\bold{jawaban \ E}}\end{align}[/tex]