Question:
If tan (A + B) = p, tan (A – B) = q, then show that tan 2A = (p + q) / (1 – pq).
[Hint: Use 2A = (A + B) + (A – B)]
Solution:
We know that,
tan 2A = tan (A + B + A – B)
And also,
$\tan (\mathrm{x}+\mathrm{y})=\frac{\tan \mathrm{x}+\tan \mathrm{y}}{1-\tan \mathrm{x} \tan y}$
$\therefore \tan 2 \mathrm{~A}=\frac{\tan (\mathrm{A}+\mathrm{B})+\tan (\mathrm{A}-\mathrm{B})}{1-\tan (\mathrm{A}+\mathrm{B}) \tan (\mathrm{A}-\mathrm{B})}$
Substituting the values given in question,
$\Rightarrow \tan 2 A=\frac{\mathrm{p}+\mathrm{q}}{1-\mathrm{pq}}$
Hence, $\tan 2 \mathrm{~A}=\frac{\mathrm{p}+\mathrm{q}}{1-\mathrm{pq}}$
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