Write the following as an algebraic expression in u, u 0.
-$\tan \left( \operatorname { arcsec } \frac { \sqrt { \mathrm { u } ^ { 2 } + 9 } } { \mathrm { u } } \right)$

Question

Write the following as an algebraic expression in u, u > 0.
-$\tan \left( \operatorname { arcsec } \frac { \sqrt { \mathrm { u } ^ { 2 } + 9 } } { \mathrm { u } } \right)$

Quizplus · Accepted Answer

The expression $\tan(\operatorname{arcsec}(\frac{\sqrt{u^2+9}}{u}))$ can be simplified by considering the definition of arcsec and the properties of right triangles. The arcsec function gives an angle whose secant (hypotenuse over adjacent side in a right triangle) is $\frac{\sqrt{u^2+9}}{u}$. In a right triangle, if the adjacent side (next to the angle) is $u$ and the hypotenuse is $\sqrt{u^2+9}$, then the opposite side (using the Pythagorean theorem) is $\sqrt{(\sqrt{u^2+9})^2 - u^2} = \sqrt{u^2+9-u^2} = 3$. Therefore, $\tan$ of that angle, which is the ratio of the opposite side to the adjacent side, is $\frac{3}{u}$.

Write the Following as an Algebraic Expression in U, U tan⁡(arcsec⁡u2+9u)\tan \left( \operatorname { arcsec } \frac { \sqrt { \mathrm { u } ^ { 2 } + 9 } } { \mathrm { u } } \right)tan(arcsecuu2+9​​)

Write the Following as an Algebraic Expression in U, U $\tan \left( \operatorname { arcsec } \frac { \sqrt { \mathrm { u } ^ { 2 } + 9 } } { \mathrm { u } } \right)$