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Question: Prove that : \({{\sin }^{-1}}\dfrac{3}{5}+{{\sin }^{-1}}\dfrac{8}{17}={{\sin }^{-1}}\dfrac{77}{85}\)...

Prove that : sin⁡−135+sin⁡−1817=sin⁡−17785{{\sin }^{-1}}\dfrac{3}{5}+{{\sin }^{-1}}\dfrac{8}{17}={{\sin }^{-1}}\dfrac{77}{85}.

Explanation

Solution

Hint: We will be using the concept of inverse trigonometric functions. We will be using the formula of sin⁡−1x+sin⁡−1y{{\sin }^{-1}}x+{{\sin }^{-1}}y.

Complete step by step answer:
Now, we have to prove that sin⁡−135+sin⁡−1817=sin⁡−17785{{\sin }^{-1}}\dfrac{3}{5}+{{\sin }^{-1}}\dfrac{8}{17}={{\sin }^{-1}}\dfrac{77}{85}.
We will be taking the left hand side of the equation and prove it to be equal to the right hand side.
Now, taking L.H.S we have,
sin⁡−135+sin⁡−1817...........(1){{\sin }^{-1}}\dfrac{3}{5}+{{\sin }^{-1}}\dfrac{8}{17}...........\left( 1 \right)
We know that sin⁡−1x+sin⁡−1y{{\sin }^{-1}}x+{{\sin }^{-1}}y is =sin⁡−1(x1−y2+y1−x2).........(2)={{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}} \right).........\left( 2 \right)
Where x,y≥0 and x2+y2≤1x,y\ge 0\ and\ {{x}^{2}}+{{y}^{2}}\le 1.
We will be using (2) to solve (1), but first we have to check whether x,y≥0 and x2+y2≤1x,y\ge 0\ and\ {{x}^{2}}+{{y}^{2}}\le 1.
Comparing (1) and (2) we have,
x=35>0 y=817>0 Also, x2+y2=925+64289 =9×289+64×2525×289 =420125×289 ≈0.5 \begin{aligned} & x=\dfrac{3}{5}>0 \\\ & y=\dfrac{8}{17}>0 \\\ & Also, \\\ & {{x}^{2}}+{{y}^{2}}=\dfrac{9}{25}+\dfrac{64}{289} \\\ & =\dfrac{9\times 289+64\times 25}{25\times 289} \\\ & =\dfrac{4201}{25\times 289} \\\ & \approx 0.5 \\\ \end{aligned}
So, this shows that equation (1) satisfy both the condition as x,y≥0 and x2+y2=0.5x,y\ge 0\ and\ {{x}^{2}}+{{y}^{2}}=0.5 is less than 1.
Now, using equation (2),
sin⁡−1(35)+sin⁡−1(817)=sin⁡−1(351−82172+8171−3252) =sin⁡−1(35225172+8171652) =sin⁡−1(35×152172+8174252) =sin⁡−1(3×155×17+817×45) =sin⁡−1(917+3285) =sin⁡−1(7785) \begin{aligned} & {{\sin }^{-1}}\left( \dfrac{3}{5} \right)+{{\sin }^{-1}}\left( \dfrac{8}{17} \right)={{\sin }^{-1}}\left( \dfrac{3}{5}\sqrt{1-\dfrac{{{8}^{2}}}{{{17}^{2}}}}+\dfrac{8}{17}\sqrt{1-\dfrac{{{3}^{2}}}{{{5}^{2}}}} \right) \\\ & ={{\sin }^{-1}}\left( \dfrac{3}{5}\sqrt{\dfrac{225}{172}}+\dfrac{8}{17}\sqrt{\dfrac{16}{{{5}^{2}}}} \right) \\\ & ={{\sin }^{-1}}\left( \dfrac{3}{5}\times \sqrt{\dfrac{{{15}^{2}}}{{{17}^{2}}}}+\dfrac{8}{17}\sqrt{\dfrac{{{4}^{2}}}{{{5}^{2}}}} \right) \\\ & ={{\sin }^{-1}}\left( \dfrac{3\times 15}{5\times 17}+\dfrac{8}{17}\times \dfrac{4}{5} \right) \\\ & ={{\sin }^{-1}}\left( \dfrac{9}{17}+\dfrac{32}{85} \right) \\\ & ={{\sin }^{-1}}\left( \dfrac{77}{85} \right) \\\ \end{aligned}
L.H.S = R.H.S
Hence Proved.

Note: These types of questions are calculation and formula based. So, remembering the formulas of trigonometric functions and checking calculations is a must.