Question

Find the value of sin (cos⁻¹$\frac{3}{5}$ + cos⁻¹$\frac{12}{13}$).

Answer 1

Answer:

sin (cos⁻¹ 3/5+ cos⁻¹12/13)=63/65

Step-by-step explanation:

As we know that

$cos^{-1}x=sin^{-1}(\sqrt{1-x^{2} } )\\\\\\cos^{-1}\frac{3}{5} =sin^{-1}(\sqrt{1-(\frac{3}{5})^{2} } )\\\\\\cos^{-1}\frac{3}{5}=sin^{-1}\frac{4}{5}$

$cos^{-1}x=sin^{-1}(\sqrt{1-x^{2} } )\\\\\\cos^{-1}\frac{12}{13} =sin^{-1}(\sqrt{1-(\frac{12}{13})^{2} } )\\\\\\cos^{-1}\frac{12}{13}=sin^{-1}\frac{5}{13}$

as we know that

$sin^{-1}x+sin^{-1}y=sin^{-1}[x\sqrt{1-y^{2} } +y\sqrt{1-x^{2} }]\\\\\\sin^{-1}\frac{4}{5} +sin^{-1}\frac{5}{13} =sin^{-1}[\frac{4}{5} \sqrt{1-(\frac{5}{13})^{2} } +\frac{5}{13}\sqrt{1-(\frac{4}{5})^{2} }]\\\\\\=sin^{-1}[\frac{4}{5} \sqrt{(\frac{169-25}{169}) } +\frac{5}{13}\sqrt{(\frac{25-16}{25}) }]\\\\\\=sin^{-1}[\frac{4}{5} (\frac{12}{13}) } +\frac{5}{13}(\frac{3}{5}) }]\\\\\\=sin^{-1}[\frac{63}{65}]\\\\so\\\\sin(sin^{-1}[\frac{63}{65}])=\frac{63}{65}$