In triangle ABC, angle C is 90 °. tgA = 2√6. Find cosA.

First way.
The tangent of the angle of a right-angled triangle is the ratio of the opposite leg to the adjacent one, then:
tgBAC = BC / AC = 2 * √6 / 1.
Let the length of the segment AC = X cm, then the length of the leg BC = 2 * √6 cm.
Using the Pythagorean theorem, we determine the length of the hypotenuse AB.
AB2 = AC2 + BC2 = X2 + 24 * X2 = 25 * X2.
AB = 5 * X cm.
Then: CosBAC = AC / AB = X / 5 * X = 1/5 = 0.2.

Second way.
We use the formula for the dependence of the tangent and cosine of an angle.
1 / Cos2A = 1 + tg2A.
Cos2A = 1 / (1 + tg2A) = 1 / (1 + 24) = 1/25.
CosBAC = 1/5 = 0.2.
Answer: The cosine of the angle BAC is 0.2.