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Question: How do you use the double angle or half angle formulas to simplify \[6{\cos ^2}x - 3\]...

How do you use the double angle or half angle formulas to simplify 6cos⁡2x−36{\cos ^2}x - 3

Explanation

Solution

Here we have to simplify the given trigonometry 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 and the half angle formula is defined as cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = {\cos ^2}x - {\sin ^2}x and cos⁡2x=1+cos⁡2x2{\cos ^2}x = \dfrac{{1 + \cos 2x}}{2}
Now consider the given trigonometric function 6cos⁡2x−36{\cos ^2}x - 3. the double angle formula is defined for cos⁡2x{\cos ^2}x is cos⁡2x+sin⁡2x=cos⁡2x\cos 2x + {\sin ^2}x = {\cos ^2}x, on substituting the formula in the given trigonometric expression we have
⇒6(cos⁡2x+sin⁡2x)−3\Rightarrow 6(\cos 2x + {\sin ^2}x) - 3
On multiplying we have
⇒6cos⁡2x+6sin⁡2x−3\Rightarrow 6\cos 2x + 6{\sin ^2}x - 3
This is the simplified form by using the double angle formula.
Now consider the given trigonometric function 6cos⁡2x−36{\cos ^2}x - 3. the half angle formula is defined for cos⁡2x{\cos ^2}x is cos⁡2x=1+cos⁡2x2{\cos ^2}x = \dfrac{{1 + \cos 2x}}{2}, on substituting the formula in the given trigonometric expression we have
⇒6(1+cos⁡2x2)−3\Rightarrow 6\left( {\dfrac{{1 + \cos 2x}}{2}} \right) - 3
On simplifying we have
⇒3(1+cos⁡2x)−3\Rightarrow 3\left( {1 + \cos 2x} \right) - 3
on multiplying we get
⇒3+3cos⁡2x−3\Rightarrow 3 + 3\cos 2x - 3
The number 3 and -3 get cancels and we have
⇒3cos⁡2x\Rightarrow 3\cos 2x

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 double angle formula cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = {\cos ^2}x - {\sin ^2}x and cos⁡2x=1+cos⁡2x2{\cos ^2}x = \dfrac{{1 + \cos 2x}}{2} , with the help of the simple arithmetic operations we have simplified the given trigonometric function.