Question

if $x=\tan {}^{-1}\frac{1}{7}andy={\tan }^{-1}\frac{1}{3},then:$

• A. $\cos 2x=\sin 4y$
• B. $\cos 4y=\cos 2x$
• C. $\cos 2y=\sin 4x$
• D. $noneofthese$

Answer 1

Answer: (A)

$\cos 2x=\sin 4y$

Given $x={\tan }^{-1}\frac{1}{7}⟹2x=2{\tan }^{-1}\frac{1}{7}={\tan }^{-1}\frac{\frac{2}{7}}{1-\frac{1}{49}}={\tan }^{-1}\frac{7}{24}={\cos }^{-1}\frac{24}{\sqrt{7^2+{24}^2}}={\cos }^{-1}\frac{24}{25}$

$y={\tan }^{-1}\frac{1}{3}⟹2y=2{\tan }^{-1}\frac{1}{3}={\tan }^{-1}\frac{\frac{2}{3}}{1-\frac{1}{9}}={\tan }^{-1}\frac{3}{4}$

$⟹4y=2{\tan }^{-1}\frac{3}{4}={\tan }^{-1}\frac{2\times \frac{3}{4}}{1-\frac{9}{16}}={\tan }^{-1}\frac{24}{7}={\sin }^{-1}\frac{24}{\sqrt{{24}^2+7^2}}={\sin }^{-1}\frac{24}{25}$

$⟹\cos 2x=\sin 4y$