Question

What are the formulae of:
(1) $ 1 + \cos \left( {2x} \right) $
(2) $ 1 - \cos \left( {2x} \right) $

Answer 1

Hint: The given problem requires us to simplify the given trigonometric expression using some simple and basic trigonometric formulae. The question describes the wide ranging applications of trigonometric identities and formulae. The question requires thorough knowledge of trigonometric functions, formulae and identities. We must keep in mind the trigonometric double angle formula of cosine in order to solve the problem. We know that there are various forms of cosine double angle formulas and we should know which one to choose in every situation so as to simplify the expression.

Complete step-by-step answer:
In the given question, we are required to find the formulae of the given two trigonometric expressions using some basic trigonometric formulae.
We must know the trigonometric double angle formulae of cosine to solve the given problem.
Now, we have to find the value of $ 1 + \cos \left( {2x} \right) $ .
So, we know that $ \cos \left( {2x} \right) = 2{\cos ^2}x - 1 $ .
Hence, substituting the value of $ \cos \left( {2x} \right) $ in $ 1 + \cos \left( {2x} \right) $ , we get,
 $ \Rightarrow 1 + \cos \left( {2x} \right) = 1 + 2{\cos ^2}x - 1 $
 $ \Rightarrow 1 + \cos \left( {2x} \right) = 2{\cos ^2}x $
So, the formula for $ 1 + \cos \left( {2x} \right) $ is $ 2{\cos ^2}x $ .
Now, we have to find the formula for $ 1 - \cos \left( {2x} \right) $
We also know that $ \cos \left( {2x} \right) = 1 - 2{\sin ^2}x $ .
So, we get,
 $ \Rightarrow 1 - \cos \left( {2x} \right) = 1 - \left( {1 - 2{{\sin }^2}x} \right) $
 $ \Rightarrow 1 - \cos \left( {2x} \right) = 1 - 1 + 2{\sin ^2}x $
 $ \Rightarrow 1 - \cos \left( {2x} \right) = 2{\sin ^2}x $
So, the formula for $ 1 - \cos \left( {2x} \right) $ is $ 2{\sin ^2}x $ .

Note: Trigonometric identities and formulae are of vital importance for solving any question involving trigonometric functions and identities. All the trigonometric ratios can be converted into each other using the simple trigonometric identities. Such questions require thorough knowledge of trigonometric conversions and formulae. Algebraic operations and simplification rules come into significant use while solving such problems. We should also exactly know when to use which trigonometric formula to simplify the trigonometric expression.