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INVERSE TRIGNOMETRIC FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINVERSE TRIGNOMETRIC FUNCTIONS5 Marks
Question
Prove that:
$\cos^{-1}\frac{4}{5}+\cos^{-1}\frac{12}{13}=\cos^{-1}\frac{33}{65}$
✓

Answer

$\text{Let}\ \cos^{-1}\frac{4}{5}=x.\text{Then}, \cos x=\frac{4}{5}$ $\Rightarrow\sin x=\sqrt{1-\bigg(\frac{4}{5}\bigg)^2}=\frac{3}{5}.$ $\therefore\tan x=\frac{3}{4}\Rightarrow x=\tan^{-1}\frac{3}{4}$ $\therefore \cos^{-1}\frac{4}{5}=\tan^{-1}\frac{3}{4} \dots\dots(1)$ Now, let $\cos^{-1}\frac{12}{13}=y.$ Then, $\cos y=\frac{12}{13}\Rightarrow\sin y=\frac{5}{13}.$ $\therefore\tan y=\frac{5}{12}\Rightarrow y=\tan^{-1}\frac{5}{12}$ $\therefore\cos^{-1}\frac{12}{13}=\tan^{-1}\frac{5}{12} \dots\dots(2)$ Let $\cos^{-1}\frac{33}{65}=z.$ Then, $\cos z=\frac{33}{65}\Rightarrow\sin z=\frac{56}{65}.$ $\therefore\tan z=\frac{56}{33}\Rightarrow z=\tan^{-1}\frac{56}{33}$ $\therefore\cos^{-1}\frac{33}{65}=\tan^{-1}\frac{56}{33} \dots\dots(3)$ Now, we will prove that: $\text{L.H.S.}=\cos^{-1}\frac{4}{5}+\cos^{-1}\frac{12}{13}$$=\tan^{-1}\frac{3}{4}+\tan^{-1}\frac{5}{12}$ [Using(1) and (2)]
$=\tan^{-1}\frac{\frac{3}{4}+\frac{5}{12}}{1-\frac{3}{4}.\frac{5}{12}}$ $ \bigg[\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}\bigg]$
$=\tan^{-1}\frac{36+20}{48-15}$
$=\tan^{-1}\frac{56}{33}$
$=\tan^{-1}\frac{56}{33}\ [\text{by}(3)]$
 $=\cos^{-1}\frac{33}{65}\ \text{R.H.S.}$

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