Question

What does `cos(arctan(pi/2))-sin(arc cot(pi/4))` equal?

Answer 1

`cos (arc tan (frac{pi }{2}))-sin (arc cot (frac{pi }{4}))=frac{2}{sqrt{pi ^2+4}}-frac{4}{sqrt{pi ^2+16}}`

Explanation:

`cos (arctan (frac{pi }{2}))-sin (arc cot (frac{pi }{4}))`
We know,
`cos (arctan (frac{pi }{2}))`
using the following identity,
`cos (arctan (x))=frac{1}{sqrt{1+x^2}}`
`=frac{1}{sqrt{1+(frac{pi }{2})^2}}`
Refining it,
`=\frac{2}{\sqrt{4+\pi ^2}}`

Also,
`sin (arc cot (frac{pi }{4}))`
using the following identity,
`sin (arc cot (x))=frac{1}{sqrt{1+x^2}}`
`sin (arc cot (x))=frac{1}{sqrt{1+x^2}}`
Refining it,
`=frac{4}{sqrt{16+pi ^2}}`

Finally,
`=frac{2}{sqrt{pi ^2+4}}-frac{4}{sqrt{pi ^2+16}}`