Question

If A = 30⁰ , verify that:

(ii) cos 2A = ${\displaystyle \frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}}$

Answer 1

A = 30⁰ 
⇒ 2A = 2 × 30⁰  = 60⁰ 

(ii) cos 2A = cos ${\displaystyle {60}^0=\frac{1}{2}}$

${\displaystyle \frac{1-\tan 62A}{1+{\tan }^{2}A}=\frac{1-{\tan }^2{30}^0}{1+{\tan }^2{30}^0}=\frac{1-{\left(\frac{1}{\sqrt{3}}\right)}^2}{1+{\left(\frac{1}{\sqrt{3}}\right)}^2}}$ = ${\displaystyle \frac{1-\left(\frac{1}{3}\right)}{1+\left(\frac{1}{3}\right)}=\frac{\left(\frac{2}{3}\right)}{\frac{4}{3}}=\left(\frac{2}{3}\right)\times \frac{3}{4}=\frac{1}{2}}$

∴ cos 2A = ${\displaystyle \frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}}$