Question

What is the value of $\cos \left( \dfrac{\pi }{7} \right)\cos \left( \dfrac{\pi }{5} \right)-\sin \left( \dfrac{\pi }{7} \right)\sin \left( \dfrac{\pi }{5} \right)$ ?

Answer 1

Hint: We can clearly see that what we are given is the trigonometric function that we need to solve directly or by simplification. We can make use of the identities in order to get the answer quickly and avoid error in substituting or calculating the values.

Complete step-by-step solution:
Trigonometric identities are useful wherever trigonometric functions are involved in an expression or equation. Identity inequalities which are true for every value occurring on both sides of an equation. The identities involve certain functions of one or more angles.
In the given expression we can see we are getting some form that we are having two different angles with sine and cosine trigonometric functions.
Now if we remember the identities of trigonometric functions then we can easily simply and directly find the answer. We know identities in different forms but in this we can see that the angle sum property of trigonometric function is needed so for that what we need to do is just recall that which is $\cos (a+b)=\cos a\cos b-\sin a\sin b$ .
Now observing the above identity of function, we can clearly see that what we are given in the question is similar to the right hand side of the given identity.
Now, comparing from the question we get a=$\dfrac{\pi }{7}$ and b= $\dfrac{\pi }{5}$ .
Already our expression is equal to right hand side of identity, so directly making it equivalent to left hand side we get $\cos \left( \dfrac{\pi }{7}+\dfrac{\pi }{5} \right)$ which is equal to $\cos \left( \dfrac{12\pi }{35} \right)$ .

Note: Try to remember the various forms of identities and learn to know where to use which identity. We must also remember the various angle related properties of trigonometric functions and their relations.