Question

# prove that sin^−1(x)=csc^−1(1/x)

Functions

prove that $$\sin^{−1}(x)=\csc^{-1}\left(\frac{1}{x}\right)$$

2021-01-09

Let $$\sin^{−1}x=y.$$

Then $$\sin y=x$$.

Cosecant and sine are reciprocals so if $$\sin y=x$$, then $$\csc y=\frac{1}{x}$$.

If $$\csc y=\frac{1}{x}$$, then $$y=\csc^{−1}\left(\frac{1}{x}\right)$$

Since $$y=\sin^{−1}\times x$$, then $$\sin^{−1}x=\csc^{−1}\left(\frac{1}{x}\right)$$ since they both equal y.