Relation Between $$\sin^{-1}x$$ and $$\tan^{-1}x$$

Using

$$\sin^{-1}\left(\frac35\right)=\tan^{-1}\left(\frac34\right),$$

we get

$$\tan^{-1}\left(\frac34\right) -2\tan^{-1}\left(\frac12\right)$$

Now,

$$2\tan^{-1}\left(\frac12\right) = \tan^{-1}\left(\frac43\right)$$

Hence,

$$\tan^{-1}\left(\frac34\right) - \tan^{-1}\left(\frac43\right)$$

Using

$$\tan^{-1}a-\tan^{-1}b = \tan^{-1}\left(\frac{a-b}{1+ab}\right),$$

$$= \tan^{-1}\left( \frac{\frac34-\frac43}{1+\frac34\cdot\frac43} \right)$$

$$= \tan^{-1}\left( \frac{-\frac7{12}}{2} \right)$$

$$= \tan^{-1}\left(-\frac7{24}\right)$$

Therefore,

$$k=-\frac7{24}$$