Question

The value of the expression $\cos {1^ \circ }.\cos {2^ \circ }.\cos {3^ \circ }........\cos {179^ \circ } =$
a) $- 1$
b) $0$
c) $\dfrac{1}{{\sqrt 2 }}$
d) $1$

Answer 1

Hint : We need to find the value of the expression $\cos {1^ \circ }.\cos {2^ \circ }.\cos {3^ \circ }........\cos {179^ \circ }$. We only know the trigonometric value of some basic measure of angles and so we cannot just find the value of each term and multiply. We simply know that $\cos {90^ \circ } = 0$and $\cos {1^ \circ }.\cos {2^ \circ }.\cos {3^ \circ }........\cos {179^ \circ }$ include $\cos {90^ \circ }$ in between. $0$ multiplied with any number or any number of terms gives $0$. So, we will find our answer using these properties

** Complete step-by-step answer** :
We need to find the value of $\cos {1^ \circ }.\cos {2^ \circ }.\cos {3^ \circ }........\cos {179^ \circ }$.
$\cos {1^ \circ }.\cos {2^ \circ }.\cos {3^ \circ }........\cos {179^ \circ } = \cos {1^ \circ } \times \cos {2^ \circ } \times \cos {3^ \circ } \times ........ \times \cos {179^ \circ }$
As the angle measures are forward counting numbers, it includes $\cos {90^ \circ }$ in between. So,
$\cos {1^ \circ }.\cos {2^ \circ }.\cos {3^ \circ }........\cos {179^ \circ } = \cos {1^ \circ } \times \cos {2^ \circ } \times \cos {3^ \circ } \times ........ \times \cos {179^ \circ }$
$= \cos {1^ \circ } \times \cos {2^ \circ } \times \cos {3^ \circ } \times .... \times \cos {90^ \circ } \times .... \times \cos {179^ \circ }$
We know the value of $\cos {90^ \circ } = 0$. Hence, putting the value of $\cos {90^ \circ }$, we get
$\cos {1^ \circ }.\cos {2^ \circ }.\cos {3^ \circ }........\cos {179^ \circ } = \cos {1^ \circ } \times \cos {2^ \circ } \times \cos {3^ \circ } \times ........ \times \cos {179^ \circ }$
$= \cos {1^ \circ } \times \cos {2^ \circ } \times \cos {3^ \circ } \times .... \times \cos {90^ \circ } \times .... \times \cos {179^ \circ }$
$= \cos {1^ \circ } \times \cos {2^ \circ } \times \cos {3^ \circ } \times .... \times 0 \times .... \times \cos {179^ \circ }$
Now, we know anything multiplied with $0$ gives $0$i.e. $a \times b \times c \times ...... \times 0 \times ..... \times n = 0$.
Hence,
$= \cos {1^ \circ } \times \cos {2^ \circ } \times \cos {3^ \circ } \times .... \times \cos {90^ \circ } \times .... \times \cos {179^ \circ }$
$= \cos {1^ \circ } \times \cos {2^ \circ } \times \cos {3^ \circ } \times .... \times 0 \times .... \times \cos {179^ \circ }$
$= 0$
So, we got
$\cos {1^ \circ }.\cos {2^ \circ }.\cos {3^ \circ }........\cos {179^ \circ } = 0$
So, the correct answer is “Option b”.

Note : In such types of questions, we usually first think of using some trigonometric identity to solve. Also, we need to observe the pattern of angle measures very carefully as many of the times it follows some pattern like here it was simple forward counting it could have been odd numbers or even numbers or anything else. And, we forget or mix up the values of trigonometric functions for the basic angles which we need to learn and revise otherwise we won't be able to think of such tricks and solve the questions easily.