Question

How do you use the double angle formula to rewrite the expression \[\dfrac{1}{3}{\cos ^2}x - \dfrac{1}{6}\]

Answer 1

Hint: Here we have to simplify the given trigonometric expression. In the question it’s already mentioned that we have to solve the above function by using the double angle or half angle formula. By using the formulas of double angle and half angle trigonometry ratios we can simplify the given question.

Complete step by step answer:
The concept known as a double angle is associated with the three common trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). Double, as the word implies, means to increase the size of the angle to twice its size.
The double angle formula is defined as \[\cos 2x = {\cos ^2}x - {\sin ^2}x\]
Now consider the given trigonometric function \[\dfrac{1}{3}{\cos ^2}x - \dfrac{1}{6}\]. the double angle formula is defined for \[{\cos ^2}x\] is \[\cos 2x + {\sin ^2}x = {\cos ^2}x\], on substituting the formula in the given trigonometric expression we have
\[ \Rightarrow \dfrac{1}{3}(\cos 2x + {\sin ^2}x) - \dfrac{1}{6}\]
On simplifying we have
\[ \Rightarrow \dfrac{{\cos 2x}}{3} + \dfrac{{{{\sin }^2}x}}{3} - \dfrac{1}{6}\]
We take the LCM for the terms we have
\[ \Rightarrow \dfrac{{2\cos 2x + 2{{\sin }^2}x - 1}}{6}\]
This is the simplified form by using the double angle formula.
Here we are not using the half angle formula to simplify the given trigonometric function. Because they have mentioned in the to use only the double angle formula. If in the question they mention to solve the half angle formula then we have to use the formula and simplify the trigonometric functions.

Note: In the question they have already mentioned to solve the given problem using the double or half angle formula. Therefore we must know about the formula. Here we have used the double angle formula \[\cos 2x = {\cos ^2}x - {\sin ^2}x\] , with the help of the simple arithmetic operations we have simplified the given trigonometric function.