Trigonometry & Inverse Trigonometry
Right triangle properties
Grade 11
Question:
<p>Let ABC be a right triangle with $\angle BAC = \frac{\pi}{2}$, then $\left(\frac{r_2}{2R^2} + \frac{r}{R}\right)$ is equal to: (where symbols used have usual meaning in a triangle)</p>
<p>(a) $\sin B \sin C$</p>
<p>(b) $\tan B \tan C$</p>
<p>(c) $\sec B \sec C$</p>
<p>(d) $\cot B \cot C$</p>
Step-by-Step Solution
Key Concept: Use special properties of right triangles where one angle is 90° to simplify exradii and inradii formulas.
<p>For a right triangle with $\angle A = 90°$, the circumradius $R = \frac{a}{2}$ where $a$ is the hypotenuse.</p><p>The inradius is $r = \frac{b+c-a}{2}$ and $r_2 = \frac{\Delta}{s-b}$.</p><p>After substitution and simplification using $B + C = 90°$, the expression equals $\cot B \cot C$.</p>
Correct Answer: D