Question

prove that sin^−1(x)=csc^−1(1/x)

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asked 2021-01-08

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

Answers (1)

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.

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