Question

If $0<{\cos }^{-1}x<1$ and $1+\sin \left({\cos }^{-1}x\right)+{\sin }^2\left({\cos }^{-1}x\right)+{\sin }^3\left({\cos }^{-1}x\right)+.........\infty =2,$ then find $2\sqrt{3}x.$

Answer 1

We have $1+sin\left(co{s}^{-1}x\right)+si{n}^2\left(co{s}^{-1}x\right)+.........\infty =2$

This is an infinite $GP$

$\Rightarrow \frac{1}{1-sin\left(co{s}^{-1}x\right)}=2\Rightarrow \frac{1}{2}=1-sin\left(co{s}^{-1}x\right)$

$\Rightarrow sin\left(co{s}^{-1}x\right)=\frac{1}{2}$

$\Rightarrow co{s}^{-1}x=\frac{\pi }{6}\therefore x=\frac{\sqrt{3}}{2}Ans.$

$\Rightarrow 2\sqrt{3}x=2\sqrt{3}\times \frac{\sqrt{3}}{2}=3$