Prove that cossin-1-913-sintan-123

L.H.S cos(sin^ 1√(913)) We know that cos(sin^ 1x)=√(1 x^2) So, cos(sin^ 1√(913))=√(1 (√(913))^2) cos(sin^ 1√(913))=√(1 (913)) cos(sin^ 1√( ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

First Principle of Differentiation

Prove that c...

Question

Prove that
$cos ({sin}^{- 1} \sqrt{\frac{9}{13}}) = sin ({tan}^{- 1} \frac{2}{3})$

Open in App

Solution

L.H.S
$cos ({sin}^{- 1} \sqrt{\frac{9}{13}})$

We know that
$cos ({sin}^{- 1} x) = \sqrt{1 - x^{2}}$

So,
$cos ({sin}^{- 1} \sqrt{\frac{9}{13}}) = \sqrt{1 - {(\sqrt{\frac{9}{13}})}^{2}}$

$cos ({sin}^{- 1} \sqrt{\frac{9}{13}}) = \sqrt{1 - (\frac{9}{13})}$

$cos ({sin}^{- 1} \sqrt{\frac{9}{13}}) = \sqrt{(\frac{4}{13})}$

$cos ({sin}^{- 1} \sqrt{\frac{9}{13}}) = \frac{2}{\sqrt{13}}$

R.H.S
$sin ({tan}^{- 1} \frac{2}{3})$

We know that
$sin ({tan}^{- 1} x) = \frac{x}{\sqrt{1 + x^{2}}}$

So,
$sin ({tan}^{- 1} \frac{2}{3}) = \frac{\frac{2}{3}}{\sqrt{1 + {(\frac{2}{3})}^{2}}}$

$sin ({tan}^{- 1} \frac{2}{3}) = \frac{\frac{2}{3}}{\sqrt{1 + \frac{4}{9}}}$

$sin ({tan}^{- 1} \frac{2}{3}) = \frac{\frac{2}{3}}{\sqrt{\frac{13}{9}}}$

$sin ({tan}^{- 1} \frac{2}{3}) = \frac{2}{\sqrt{13}}$

Hence, proved.

Suggest Corrections

Similar questions

Q. Show that

cot ({sin}^{- 1} \sqrt{\frac{13}{17}}) = sin ({tan}^{- 1} \frac{2}{3})

Q. Evaluate the following:
(i)

\cos (\sin^{- 1} \frac{3}{5})

(ii)

\sin (\cos^{- 1} \frac{4}{5})

(iii)

\cos (\sin^{- 1} - \frac{3}{5})

(iv)

\tan (\cos^{- 1} \frac{8}{17})

(v)

cosec \{\cos^{- 1} (- \frac{12}{13})\}

(vi)

\tan \{2 \tan^{- 1} \frac{1}{5} - \frac{π}{4}\}

(vii)

\tan \frac{1}{2} (\cos^{- 1} \frac{\sqrt{5}}{3})

(viii)

\sin (\frac{1}{2} \cos^{- 1} \frac{4}{5})

(ix)

\cos (\sin^{- 1} \frac{3}{5} + \sin^{- 1} \frac{5}{13})

(x) sin (tan⁻¹ x + cot⁻¹ x)

Q. Prove the following:

cos ({sin}^{- 1} \frac{3}{5} + {cot}^{- 1} \frac{3}{2}) = \frac{6}{5 \sqrt{13}} .

Q. Prove that:

{tan}^{- 1} \frac{1}{4} + {tan}^{- 1} \frac{2}{9} = \frac{1}{2} {cos}^{- 1} \frac{3}{5}

Prove that:
(i) $\frac{1}{3 + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + 1} = 1$
(ii) $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}} = 2$

Join BYJU'S Learning Program

Prove thatcos(sin−1√913)=sin(tan−123)

Prove that
$cos ({sin}^{- 1} \sqrt{\frac{9}{13}}) = sin ({tan}^{- 1} \frac{2}{3})$