Question

Triangle XYZ is an isosceles right triangle, right angled at X. Find the value of cosec² Y + tan² Z.

Answer 1

Since ΔXYZ is right angled at X and is isosceles, the other two angles Y and Z are equal.

Sum of angles in a triangle = 180°

∠X = 90°, so ∠Y + ∠Z = 90°

Since ∠Y = ∠Z, each is 45°

Calculate:

cosec² Y + tan² Z = cosec² 45° + tan² 45°

`"cosec"  45^circ = 1/(sin 45^circ)`

= `1/(sqrt(2)/2`

= `sqrt(2)`

`"cosec"^2  45^circ = (sqrt(2))^2` = 2

tan 45° = 1

tan² 45° = 1² = 1

So, cosec² Y + tan² Z = 2 + 1 = 3.