Given that 0 ≤ 𝑥 ≤ 𝜋/2and 0 ≤ 𝑦 ≤𝜋/2

Question

find the exact value of 𝑡𝑎𝑛(𝑥 + 𝑦) if 𝑐𝑜𝑡(𝑥) =6/5 and 𝑠𝑒𝑐(𝑦) =3/2

Answer 1

cotx = 6/5 ---> tanx = 5/6
secy = 3/2 ---> cosy = 2/3

sketch your triangles in quad I
using Pythagoras,
if tanx = 5/6
r^2 = 25 + 36 = 61
r = √61

if cosy = 2/3
by Pythagoras: 2^2 + Y^2 = 3^2
Y = √5
tany = √5/3

tan(x+y) = (tanx + tany)/(1 - tanxtany)
= (5/6 + √5/3)/(1 - (5/6)(√5/3)
= ( (5 + 2√5)/6 )/ ( (6 - 5√5)/6)
= (5 + 2√5)/(6 - 5√5)