Question

For questions 1 – 2, find an algebraic expression equivalent to the following expressions:

1. cos(tan^-1x)

2. cot(arcsin x)

Answer 1

Let
`y=\tan^(-1)x`, so that
`\tan y=x`. Picture a right triangle with hypotenuse of length
`z` and a reference angle
`y`. Then

`\cos y=\frac 1z`

We have

`\tan^2y=\sec^2y-1\implies x^2=z^2-1\implies z=√(x^2+1)`

`\implies\cos(\tan^(-1)x)=\cos y=\frac1{√(x^2+1)}`

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Similar strategy: Let
`y=\arcsin x`, so that
`\sin y=x`. In a right triangle with hypotenuse
`z` and reference angle
`y`, we have

`\cos y=√(1-x^2)[t/ex]</p><p>[tex]\implies\cot(\arcsin x)=\cot y=(\cos y)/(\sin y)=\frac{√(1-x^2)}x`