Question

The expression $2\cos \dfrac{\pi }{13}\cos \dfrac{9\pi }{13}+\cos \dfrac{3\pi }{13}+\cos \dfrac{5\pi }{13}$ is equal to
1). $-1$
2). $0$
3). $1$
4). None of these

Answer 1

To solve this problem, first we need to understand the cosine function of the trigonometry and then we will observe the given question and apply some trigonometric functions to make it simpler and then apply some cosine properties and then you will get the required answer.

Complete step-by-step solution:
Trigonometry can be defined as a study of the relationship of angles, lengths and heights. There are a total six types of different functions in trigonometry: Sine, Cosine, Secant, Cosecant, Tangent and Cotangent. Basically these six types of trigonometric functions define the relationship between the different sides of a right angle triangle. We can define six functions as:
Sine is defined as the ratio of opposite sides to the hypotenuse.
Tangent can be defined as the ratio of opposite sides to the adjacent side.
Cotangent is the reciprocal of the tangent. So it is the ratio of adjacent sides to the opposite side.
Cosecant is the reciprocal of sine. So it can be defined as the ratio of hypotenuse to the opposite side.
Secant is the reciprocal of the cosine. So it can be defined as the ratio of hypotenuse to adjacent side.
Cosine is defined as the ratio between the adjacent side of the right angled triangle and the hypotenuse of the right angled triangle. It is one of the three main primary trigonometric functions. It is interesting to know that the values of cosine change according to the quadrants. As cosine has positive value in first quadrant and fourth quadrant and negative value in second quadrant and third quadrant.
Cosine function can also be defined as the complement of the sine function. The graph of cosine function is an up-down graph just like the sine graph. The only difference between the sine graph and the cosine graph is that the sine graph starts from $0$ while the cosine graph starts from $90$ .
As, we are given in the question:
$2\cos \dfrac{\pi }{13}\cos \dfrac{9\pi }{13}+\cos \dfrac{3\pi }{13}+\cos \dfrac{5\pi }{13}$
Now, we will use the formula $2\cos A\cos B=\cos (A+B)+\cos (A-B)$
$= \cos \left( \dfrac{\pi }{13}+\dfrac{9\pi }{13} \right)+\cos \left( \dfrac{\pi }{13}-\dfrac{9\pi }{13} \right)+\cos \dfrac{3\pi }{13}+\cos \dfrac{5\pi }{13}$
$= \cos \dfrac{10\pi }{13}+\cos \left( -\dfrac{8\pi }{13} \right)+\cos \dfrac{3\pi }{13}+\cos \dfrac{5\pi }{13}$
We will use this property $\cos (-\theta )=\cos \theta$
$= \cos \dfrac{10\pi }{13}+\cos \dfrac{8\pi }{13}+\cos \dfrac{3\pi }{13}+\cos \dfrac{5\pi }{13}$
$= \cos \left( \pi -\dfrac{3\pi }{13} \right)+\cos \left( \pi -\dfrac{5\pi }{13} \right)+\cos \dfrac{3\pi }{13}+\cos \dfrac{5\pi }{13}$
Now, as we know $\cos (\pi -\theta )=-\cos \theta$
$= -\cos \dfrac{3\pi }{13}-\cos \dfrac{5\pi }{13}+\cos \dfrac{3\pi }{13}+\cos \dfrac{5\pi }{13}$
$= 0$
Hence, the correct option is $2$.

Note: Trigonometry is used in various things in day to day life, such as measuring fields and areas, making walls parallel and perpendicular, in roof inclination, architects use trigonometry to calculate structural load, and in knowing sun shading and light angles, and also used in the components of vectors.