Question

If $p = \cos 55^\circ$, $q = \cos 65^\circ$, $r = \cos 175^\circ$ then the value of $\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{{pq}}$ is

Answer 1

Here, we will first take the LCM of the given expression. Then we will substitute the given values in the expression. We will then simplify the expression using trigonometric formulas and identities. We will solve the equation further to get the required value.

Formula Used:
$\cos A + \cos B = 2\cos \left( {\dfrac{{A + B}}{2}} \right)\cos \left( {\dfrac{{A - B}}{2}} \right)$

Complete step-by-step answer:
It is given that $p = \cos 55^\circ$, $q = \cos 65^\circ$ and $r = \cos 175^\circ$.
First, we will take the LCM of the denominators of the given expression $\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{{pq}}$, we get
$\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{{pq}} = \dfrac{{q + p + r}}{{pq}}$
Now, substituting the given values in this fraction, we get,
$\Rightarrow \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{{pq}} = \dfrac{{\cos 65^\circ + \cos 55^\circ + \cos 175^\circ }}{{\cos 55^\circ \times \cos 65^\circ }}$
Now, using the formula $\cos A + \cos B = 2\cos \left( {\dfrac{{A + B}}{2}} \right)\cos \left( {\dfrac{{A - B}}{2}} \right)$ in the numerator for $\cos 65^\circ + \cos 55^\circ$, we get,
$\Rightarrow \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{{pq}} = \dfrac{{2\cos 60^\circ \cos 5^\circ + \cos \left( {180^\circ - 5^\circ } \right)}}{{\cos 55^\circ \times \cos 65^\circ }}$
Now, as we know, $\cos \left( {180^\circ - \theta } \right) = - \cos \theta$ because in the second quadrant, cosine is negative. Therefore, we get
$\Rightarrow \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{{pq}} = \dfrac{{2\cos 60^\circ \cos 5^\circ - \cos 5^\circ }}{{\cos 55^\circ \times \cos 65^\circ }}$
Now, substituting $\cos 60^\circ = \dfrac{1}{2}$ in the above equation, we get
$\Rightarrow \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{{pq}} = \dfrac{{2 \times \dfrac{1}{2} \times \cos 5^\circ - \cos 5^\circ }}{{\cos 55^\circ \times \cos 65^\circ }}$
$\Rightarrow \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{{pq}} = \dfrac{{\cos 5^\circ - \cos 5^\circ }}{{\cos 55^\circ \times \cos 65^\circ }}$
Subtracting the terms in the numerator, we get
$\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{{pq}} = \dfrac{0}{{\cos 55^\circ \times \cos 65^\circ }} = 0$
Therefore, the value of $\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{{pq}}$ is 0.

Note: Trigonometry is a branch of mathematics that helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers (to make maps). It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’ and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.