Question

Write the given expression in terms of x and y only tan(sin−1 x + cos−1 y)

Answer 1

`\tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )=(xy+√(\left ( 1-x^2 \right )\left ( 1-y^2 \right )))/(y√(1-x^2)-x√(1-y^2))`

Explanation:

We need to express
`\tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )` in terms of x and y .

Let
`\sin ^(-1)x=\theta \,,\,\cos ^(-1)y=\phi` , we get

`x=\sin \theta \,,\,y=\cos \phi`

Formulae Used:

`\sin ^2\theta +\cos ^2\theta =1\\\sin ^2 \phi +\cos ^2 \phi =1`

`\cos \theta =√(1-\sin^2 \theta)=√(1-x^2)\\\sin \phi=√(1-\cos ^2 \phi)=√(1-y^2)`

`\tan \theta =(\sin \theta )/(\cos\theta )=(x)/(√(1-x^2))\\\tan \phi =(\sin \phi)/(\cos\phi)=(√(1-y^2))/(y)`

We know that
`\tan \left ( \theta +\phi\right )=(\tan \theta +\tan \phi )/(1-\tan \theta \,\tan \phi )`

`\tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )=((x)/(√(1-x^2))+(√(1-y^2))/(y))/(1-(x√(1-y^2))/(y√(-x^2)))\\\therefore \tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )=(xy+√(\left ( 1-x^2 \right )\left ( 1-y^2 \right )))/(y√(1-x^2)-x√(1-y^2))`