Question

If $\tan A + \sec A = 4$, how do you find the value of $\cos A$?

Answer 1

Hint: To solve these questions, first convert the trigonometric ratios given in the question to the ratio required in the solution. Then, after simplifying, apply trigonometric identities to the solution to get the required answer.

Formula Used: The following algebraic and trigonometric formulae can be applied in this question-
$\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)$
${\sec ^2}\theta = 1 + {\tan ^2}\theta$

Complete step by step answer:
Given, $\tan A + \sec A = 4$ $......\left( i \right)$
Since, we know the trigonometric identity,
${\sec ^2}A = 1 + {\tan ^2}A$
$\Rightarrow {\sec ^2}A - {\tan ^2}A = 1$
Therefore, from the above step we get,
$\Rightarrow \left( {\sec A + \tan A} \right)\left( {\sec A - \tan A} \right) = 1$
Now, substitute the value of $\tan A + \sec A = 4$ from equation $\left( i \right)$
$\Rightarrow \left( 4 \right)\left( {\sec A - \tan A} \right) = 1$ $\left( {from\left( i \right)} \right)$
Taking $4$ to the denominator of L.H.S. from R.H.S. , we get,
$\Rightarrow \sec A - \tan A = \dfrac{1}{4}$ $.....\left( {ii} \right)$
Now, add the equations $(i)$ and $(ii)$ to further get,
$\Rightarrow \sec A + \tan A + \sec A - \tan A = 4 + \dfrac{1}{4}$
Cancel the terms that are opposite in signs, in this case, it is $\tan A$ , and take the L.C.M. on L.H.S to further simplify the expression as,
$\Rightarrow 2\sec A = \dfrac{{17}}{4}$
Take $2$ to the denominator of the L.H.S. and multiply it with $4$ to get,
$\Rightarrow \sec A = \dfrac{{17}}{8}$
Since $\sec A$ is the reciprocal of $\cos A$ that is, $(\sec A = \dfrac{1}{{\cos A}})$ therefore,
$\Rightarrow \dfrac{1}{{\cos A}} = \dfrac{{17}}{8}$
 Cross multiply the above fractions to get,
$\Rightarrow \cos A = \frac{8}{{17}}$

Therefore, if the value of $\tan A + \sec A = 4$, the value of $\cos A = \dfrac{8}{{17}}$.

Additional Information:
These questions can also be solved by converting the given trigonometric ratios in the question to simpler ratios and then by using different trigonometric identities to further simplify the solution. Such questions can be solved in more than one way, by manipulating different trigonometric identities according to the question.

Note: Keep in mind to always check which trigonometric ratio is required in the solution and then convert the trigonometric ratios given in the question accordingly. Also, some common algebraic identities like, $\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)$ can prove to be very helpful while solving these types of questions.