Find an algebraic expression equivalent to the given expression.
-$\sec \left( \operatorname { arccot } \frac { \sqrt { 5 } } { u } \right)$
A) $\frac { \sqrt { 5 u ^ { 2 } + 25 } } { 5 }$
B) $\frac { \sqrt { 5 } } { u }$
C) $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 5 } } { \mathrm { u } ^ { 2 } + 5 }$
D) $\frac { \sqrt { \mathrm { u } ^ { 2 } + 5 } } { 5 }$

Question

Quizplus · Accepted Answer

To find an algebraic expression equivalent to $\sec(\arccot(\frac{\sqrt{5}}{u}))$, we first consider the trigonometric identity involving secant and cotangent, which is $\sec(\theta) = \frac{1}{\cos(\theta)}$ and $\cot(\theta) = \frac{1}{\tan(\theta)}$. The expression inside the secant function is the arccotangent of $\frac{\sqrt{5}}{u}$, which represents an angle whose cotangent is $\frac{\sqrt{5}}{u}$. To find the secant of this angle, we need to find the hypotenuse over the adjacent side in a right triangle where the opposite side over the adjacent side is $u$ over $\sqrt{5}$. Using the Pythagorean theorem, the hypotenuse is $\sqrt{u^2 + 5}$, making the secant of the angle $\frac{\sqrt{u^2 + 5}}{\sqrt{5}}$, which simplifies to $\frac{\sqrt{5u^2 + 25}}{5}$ when rationalized.

Find an Algebraic Expression Equivalent to the Given Expression sec⁡(arccot⁡5u)\sec \left( \operatorname { arccot } \frac { \sqrt { 5 } } { u } \right)sec(arccotu5​​)

Find an Algebraic Expression Equivalent to the Given Expression $\sec \left( \operatorname { arccot } \frac { \sqrt { 5 } } { u } \right)$