If cos A = 2/5, find the value of 4 + 4tan²A.

Cosine θ is given by,

Cos θ = …(i)

Comparing cos A = 2/5 by equation (i), we get

=

Thus, we have base of triangle and hypotenuse as 2x and 5x repectively.

We know that, tan θ =

We have

We need to find hypotenuse for tangent.

Using Pythogoras Theorem,

(hypotenuse)² = (perpendicular)² + (base)²

⇒ (5x)² = (perpendicular)² + (2x)²

⇒ 25x² = (perpendicular)² + 4x²

⇒ (perpendicular)² = 25x² – 4x²

⇒ (perpendicular)² = 21x²

⇒ perpendicular = √(21x²)

⇒ perpendicular = √21 x

So tan A =

⇒ tan² A =

To solve (4 + 4tan²A), substitute value of tan² A in here.

We get

4 + 4tan²A = 4 + 4(21/4)

= 4 + 21

= 25

Hence, the value of 4 + 4tan²A is 25.