Question

proved this . chapter : inverse trigonometric functions​

Answer 1

Topic :-

Inverse Trigonometric Function

To Prove :-

$\mathtt{4\tan^{-1}\dfrac{1}{5}-\tan^{-1}\dfrac{1}{239}=\dfrac{\pi}{4}}$

Formula to be Used :-

$\mathtt{\tan^{-1}x+\tan^{-1}y=\tan^{-1}\dfrac{x+y}{1-xy}}$

where x > 0, y > 0 and xy < 1.

$\mathtt{\tan^{-1}x-\tan^{-1}y=\tan^{-1}\dfrac{x-y}{1+xy}}$

where x > 0 and y > 0.

Solution :-

Solving LHS,

$\mathtt{4\tan^{-1}\dfrac{1}{5}-\tan^{-1}\dfrac{1}{239}}$

$\mathtt{2\left (\tan^{-1}\dfrac{1}{5}+\tan^{-1}\dfrac{1}{5}\right )-\tan^{-1}\dfrac{1}{239}}$

$\mathtt{2\left (\tan^{-1}\dfrac{\dfrac{1}{5}+\dfrac{1}{5}}{1-\dfrac{1}{5}.\dfrac{1}{5}}\right )-\tan^{-1}\dfrac{1}{239}}$

$\mathtt{2\left (\tan^{-1}\dfrac{\dfrac{2}{5}}{\dfrac{24}{25}}\right )-\tan^{-1}\dfrac{1}{239}}$

$\mathtt{2\left (\tan^{-1}\dfrac{5}{12}\right )-\tan^{-1}\dfrac{1}{239}}$

Similarly,

$\mathtt{\tan^{-1}\dfrac{120}{119}-\tan^{-1}\dfrac{1}{239}}$

$\mathtt{\tan^{-1}\dfrac{\dfrac{120}{119}-\dfrac{1}{239}}{1+\dfrac{120}{119}.\dfrac{1}{239}}}$

$\mathtt{\tan^{-1}\dfrac{120 \times 239 - 119}{119\times 239+120}}$

$\mathtt{\tan^{-1}\dfrac{28561}{28561}}$

$\mathtt{tan^{-1}\:1}$

$\mathtt{\dfrac{\pi}{4}}$

RHS

$\mathtt{\dfrac{\pi}{4}}$

We observe that LHS = RHS.

Hence, Proved !!