+166 Let \(A\) be an acute angle such that \(tan \text{ } A + sec \text{ } A = 2.\) Find \(cos \text{ } A.\)
+129852 tan A + sec A = 2
sin A / cos A + 1/cos A = 2 multiply both sides by cos A
sin A + 1 = 2cosA square both sides
sin^2A + 2sin A + 1 = 4cos^2A
sin^2A + 2sinA + 1 = 4 ( 1 - sin^2A)
sin^2A + 2sin A + 1 = 4 - 4sin^2 A
5sin^2 A + 2sin A - 3 = 0 factor as
(5sin A - 3 ) (sin A + 1 ) = 0
So either sin A + 1 = 0 ⇒ sin A = - 1 ⇒ A = 3pi/2 not acute
Or
5sin A - 3 = 0
sin A = 3/5
So we have a 3 - 4 - 5 right triangle
So
cos A = 4/5
Proof
tanA + sec A = 3/4 + 5/4 = 8/4 = 2
