Question

. Evaluate tan(α + β) = sin(????+????) / cos(????+????) to show tan(???? + ????) = tan(????)+tan(????????) / 1−tan(????) tan(????
. Use the resulting formula to show that
tan(2????) = 2 tan(????) / 1−tan2(????)

Answer 1

Answer with Step-by-step explanation:

We are given that

$tan(\alpha+\beta)$

$\frac{sin(\alpha+\beta}{cos(\alpha+\beta)}$

By using formula$tan x=\frac{sin x}{cos x}$

$\frac{sin\alpha cos\beta+sin\beta cos\alpha}{cos\alpha cos\beta-sin\alpha sin\beta}$

By using property:$sin(x+y)=sin x cosy+cos x sin y$

$cos(x+y)=cos x cosy-sin x siny$

Divide numerator and denominator by $cos\alpha cos\beta$

Then, we get

$\frac{\frac{sin\alpha}{cos\alpha}+\frac{sin\beta}{cos\beta}}{1-\frac{sin\alpha sin\beta}{cos\alpha cos\beta}}$

$tan(\alpha+\beta)=\frac{tan\alpha+tan\beta}{1-tan\alpha tan\beta}$

Hence, proved

Substitute $\alpha=\beta$

Then we get

$tan 2\alpha=\frac{tan\alpha+tan\alpha}{1-tan^2\alpha}$

$tan(2\alpha)=\frac{2tan\alpha}{1-tan^2\alpha}$

Hence, proved.