Question

If $\cos A=\dfrac{2}{5}$, find the value of $4+4{{\tan }^{2}}A$.

Answer 1

We first convert $\cos A$ into $\sec A$ by using inverse trigonometric theorem. Then to simplify the term $4+4{{\tan }^{2}}A$, we use the formula $1+{{\tan }^{2}}A={{\sec }^{2}}A$. In the equation we put the value of $\sec A$ to find the solution.

Complete step by step answer:
We are going to use some trigonometric formula to find the solution.
We know that $1+{{\tan }^{2}}A={{\sec }^{2}}A$. Also, we have $\sec A=\dfrac{1}{\cos A}$.
It’s given that $\cos A=\dfrac{2}{5}$. We take inverse and find that $\sec A=\dfrac{1}{\cos A}=\dfrac{1}{\dfrac{2}{5}}=\dfrac{5}{2}$.
We need to find the value of $4+4{{\tan }^{2}}A$. We perform binary operations on the equation to find its value with respect to $\sec A$.
$4+4{{\tan }^{2}}A=4\left( 1+{{\tan }^{2}}A \right)=4{{\sec }^{2}}A$.
We have the value of $\sec A$. We use the value to find the solution.
So, $4{{\sec }^{2}}A=4{{\left( \sec A \right)}^{2}}=4{{\left( \dfrac{5}{2} \right)}^{2}}=\dfrac{4\times 25}{4}=25$

So, the value of $4+4{{\tan }^{2}}A$ is 25.

Note: As there is no particular given domain, we can easily find the inverse of the $\cos A$ assuming the angle A resides in the first quadrant. The general rule is followed when no particular value is mentioned. Also, in cases cos value always remains positive in the first quadrant. So, the inverse should exist there.