Question

How do you write an algebraic expression that is equivalent to `csc(arctan(x/sqrt2))`?

Answer 1

`csc(arc tan (x/sqrt2))=(sqrt2+x^2)/(+-x), x!=0.`

Explanation:

`arc tan(x/sqrt2)= theta rArr tan theta=x/sqrt2 rArr cottheta=sqrt2/x, x!=0.`

`rArr csc^2theta=1+cot^2theta=1+2/x^2=(x^2+2)/x^2.`

`rArr csc theta=sqrt(2+x^2)/(+-x)," i.e., to say,"`

`csc(arc tan (x/sqrt2))=sqrt(2+x^2)/(+-x), x!=0.`