Question

Prove that sin⁻¹$\frac{3}{5}$ + cos⁻¹$\frac{12}{13}$ = cos⁻¹$\frac{33}{65}$.

Answer 1

Answer:

Step-by-step explanation:

LHS

As we know that

$sin^{-1}x=cos^{-1}\sqrt{1-x^{2}} \\\\sin^{-1}\frac{3}{5} =cos^{-1}\sqrt{1-(\frac{3}{5})^{2}} \\\\=cos^{-1}\sqrt{1-(\frac{9}{25})} \\\\=cos^{-1}\sqrt{(\frac{25-9}{25})}\\\\=cos^{-1}\frac{4}{5}$

Now

$cos^{-1}x+cos^{-1}y=cos^{-1}[xy-\sqrt{1-x^{2}}\sqrt{1-y^{2} } }] \\\\cos^{-1}\frac{4}{5}+cos^{-1}\frac{12}{13}=cos^{-1}[\frac{4}{5}\frac{12}{13}-\sqrt{1-(\frac{4}{5})^{2}}\sqrt{1-(\frac{12}{13})^{2} } }\:] \\\\=cos^{-1}[\frac{48}{65}-\sqrt{(\frac{25-16}{25})}\sqrt{(\frac{169-144}{169}) } }\:] \\\\=cos^{-1}[\frac{48}{65}-(\frac{3}{5}){(\frac{5}{13}) } \:] \\\\=cos^{-1}[\frac{48}{65}-(\frac{3}{13})]\\\\\\=cos^{-1}[\frac{48}{65}-\frac{3}{13}]\\\\\\=cos^{-1}[\frac{48-15}{65}]$

$=cos^{-1}[\frac{33}{65}]\\\\=RHS$

hence proved