use special right triangles to evaluate the expression. \n\frac{cot^{2}\frac{pi}{4}+sec^{2}\frac{pi}{3}}{csc^…

Explanation:

Step1: Recall trig - values from special right - triangles

Recall the following trigonometric values from special right - triangles: For $\theta=\frac{\pi}{4}$ (45 - 45 - 90 triangle), $\cot\frac{\pi}{4}=1$. For $\theta = \frac{\pi}{3}$ (30 - 60 - 90 triangle), $\sec\frac{\pi}{3}=2$. For $\theta=\frac{\pi}{6}$ (30 - 60 - 90 triangle), $\csc\frac{\pi}{6}=2$.

Step2: Substitute the values into the expression

Substitute the values into the expression $\frac{\cot^{2}\frac{\pi}{4}+\sec^{2}\frac{\pi}{3}}{\csc^{2}\frac{\pi}{6}}$. We get $\frac{1^{2}+2^{2}}{2^{2}}$.

Step3: Simplify the numerator and denominator

First, simplify the numerator: $1^{2}+2^{2}=1 + 4=5$. The denominator is $2^{2}=4$. So the value of the expression is $\frac{5}{4}$.

Answer:

$\frac{5}{4}$