prove that `sec^(2)(tan^(-1)3)+cosec^2(cot^(-1)4)=27`

Answer

6

Explanation

Given `s - a = 5, s -b =3, s-c =2`
`rArr s= 10, a = 5, b = 7, and c = 8`
Now, Length of tangent from A to incircle
= Semiperimeter of `DeltaAPQ`
`5 = ((calpha)/(a) + alpha + (b alpha)/(a))/(2)`
`rArr alpha = (10a)/(a+b+c) = (5)/(2) " " [ :' Delta APQ and DeltaABC " are similar"]`
Similarly, `beta = (6b)/(a+b+c) = (21)/(10)`
and `gamma = (4c)/(a+b+c) = (8)/(5)`
`rArr alpha + beta + gamma + (62)/(10)`