Express Cosα +cosβ+cosγ+cos(α+β+γ) as the product of three cosines
Express Cosα +cosβ+cosγ+cos(α+β+γ) as the product of three cosines
$Cos\alpha + cos\beta + cos\gamma + cos(\alpha + \beta + \gamma )$
= $2cos\frac{\alpha + \beta }{2}\cos \frac{\alpha – \beta }{2} + 2cos\frac{\gamma + \alpha + \beta + \gamma }{2}cos\frac{\gamma – \alpha – \beta – \gamma }{2}$
= $2cos\frac{\alpha + \beta }{2}\cos \frac{\alpha – \beta }{2} + 2cos\frac{\alpha + \beta + 2\gamma }{2}cos\left( – \frac{\alpha + \beta }{2}\right)$
= $2cos\frac{\alpha + \beta }{2}\cos \frac{\alpha – \beta }{2} + 2cos\frac{\alpha + \beta + 2\gamma }{2}cos\frac{\alpha + \beta }{2}$
= $2cos\frac{\alpha + \beta }{2}\left(\cos \frac{\alpha – \beta }{2} + cos\frac{\alpha + \beta + 2\gamma }{2}\right)$
= $2cos\frac{\alpha + \beta }{2}\left(2cos\frac{\frac{\alpha – \beta }{2} + \frac{\alpha + \beta + 2\gamma }{2}}{2}cos\frac{\frac{\alpha – \beta }{2} – \frac{\alpha + \beta + 2\gamma }{2}}{2}\right)$
= $2cos\frac{\alpha + \beta }{2}\left(2cos\frac{\alpha – \beta + \alpha + \beta + 2\gamma }{4}cos\frac{\alpha – \beta – \alpha – \beta – 2\gamma }{4}\right)$
=$2cos\frac{\alpha + \beta }{2}\left(2cos\frac{2\alpha + 2\gamma }{4}cos\frac{ – 2\beta – 2\gamma }{4}\right)$
= $2cos\frac{\alpha + \beta }{2}\left(2cos\frac{\alpha + \gamma }{2}\cos \frac{\beta + \gamma }{2}\right)$
= $4cos\frac{\alpha + \beta }{2}cos\frac{\beta + \gamma }{2}cos\frac{\gamma + \alpha }{2}$
∴ Cosα +cosβ+cosγ+cos(α+β+γ) = $4cos\frac{\alpha + \beta }{2}cos\frac{\beta + \gamma }{2}cos\frac{\gamma + \alpha }{2}$ [Proved]