Question

Prove that
${ \cos }^{ - 1} \frac{12}{13} + \sin {}^{ - 1} \frac{3}{5} = \sin {}^{ - 1} \frac{56}{65}$
_________Pls solve fast...​

Answer 1

${\huge{\underline{\underline{\rm{\bold{SolutiOn:}}}}}}$

$\cos {}^{ - 1} \frac{12}{13} + \sin {}^{ - 1} \frac{3}{5} = \sin {}^{ - 1} \frac{56}{65}$

LHS:

$\cos {}^{ - 1} \frac{12}{13} + \sin {}^{ - 1} \frac{3}{5} \\</p><p></p><p></p><p>\sin {}^{ - 1} \sqrt{1 - (\frac{12}{3} ) {}^{2} } + \sin {}^{ - 1} \frac{3}{5} \\ </p><p> = \sin {}^{ - 1} \frac{5}{3} \sqrt{1 - ( \frac{3}{5}) {}^{2} } + \frac{3}{5} \sqrt{1 - ( \frac{5}{13} ) {}^{2} }$

$=\sin {}^{ - 1} (\frac{5}{13} \sqrt{ 1 - \frac{9}{25} } + \frac{3}{5} \sqrt{1 - \frac{25}{169} } )$

$= \sin {}^{ - 1} ( \frac{5}{13} \sqrt{ \frac{25 - 9}{9} } + \frac{3}{5} \sqrt{ \frac{169- 25}{169} } ) \\ = \sin {}^{ - 1} ( \frac{5}{13} \times \sqrt{ \frac{16}{25} } + \frac{3}{5} \times \sqrt{ \frac{144}{169} } )$

$= \sin {}^{ - 1} (\frac{5}{13} \times \frac{4}{5} + \frac{3}{5} \times \frac{12}{13} ) \\ = \sin {}^{ - 1} (\frac{20}{65} + \frac{36}{65} ) \\ = \sin {}^{ - 1} \frac{56}{65}$

= RHS.

====================proved....

${\huge{\bf{\green{Formulae\: to\: be\: used:-}}}}$

$༒\cos {}^{ - 1} x = \sin {}^{ - 1} \sqrt{1 - {x}^{2} } \\༒ \sin {}^{ - 1} x + \sin {}^{ - 1} y = \sin {}^{ - 1} (x \sqrt{1 - {y}^{2} } + y \sqrt{1 - {x}^{2} } )$