Question

In a $\Delta$ ABC,$\dfrac{\cos C+\cos A}{c+a}+\dfrac{\cos B}{b}$ is equal to [EAMCET 2001]
A. $\dfrac{1}{a}$
B. $\dfrac{1}{b}$
C. $\dfrac{1}{c}$
D. $\dfrac{c+a}{b}$

Answer 1

Hint:
This type of question can be solved using the projection formula. Firstly, cross-multiply the terms or use LCM to simplify the provided expression. The answer can then be obtained by using the projection rule. Finally, compare the outcome to the other options.

Formula Used:
The following are a few projection formulas of the triangle ABC:
$a = b \cos C + c \cos B$
$b = c \cos A + a \cos C$
$c = a \cos B + b \cos A$

Complete step-by-step solution:
We have, $\dfrac{\cos C+\cos A}{c+a}+\dfrac{\cos B}{b}$
Simplifying the expression by taking LCM
$=\dfrac{b \cos C+b \cos A+(c+a) \cos B}{b(c+a)}\\
=\dfrac{(b\cos C+c\cos B)+(b\cos A+a\cos B)}{b(c+a)}$
Using the projection formula we get;
$[\therefore a = b \cos C + c \cos B\\
\therefore b = c \cos A + a \cos C]\\
=\dfrac{a+c}{b(c+a)}\\
=\dfrac{1}{b}$
Hence, $\dfrac{\cos C+\cos A}{c+a}+\dfrac{\cos B}{b} =\dfrac{1}{b}$

So, option B is correct.

Note:
In such a question students usually use the wrong formula or misunderstand the concept of the projection. According to projection formulas, the sum of the projections of the other two sides on a triangle's other two sides determines the length of any one of its sides. This question requires knowledge of basic trigonometry. In order to get the right answer, apply the formula accordingly.