Question

Write the trigonometric expression as an algebraic expression containing u and y

$\sin \left(\sin {}^{-1}u-\cos {}^{-1}v\right)$

Short answer

$\sin \left({\sin }^{-1}u-{\cos }^{-1}v\right)=uv+\left(\sqrt{1-u^2}\right)\left(\sqrt{1-v^2}\right)$

Step-by-step solution

Step 1. The given information we have is

a trigonometric equation$\sin \left({\sin }^{-1}u-{\cos }^{-1}v\right)$

Step 2.putting sin-1u=xalso puttingsrc="data:image/svg+xml;base64,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" role="math" localid="1646478862288" cos-1v=y

We know that

$\cos x=\sqrt{1-{\sin }^{2}x}$

substituting we get

$\cos x=\sqrt{1-u^2}$

similarly we get

$\sin y=\sqrt{1-v^2}$

Step 3.  Using the formula sinx-y=sinxcosy-cosxsiny

Now, we get

$\sin \left(x-y\right)=uv+\left(\sqrt{1-u^2}\right)\left(\sqrt{1-v^2}\right)$