Question

What are the three double angle formulas for $\cos 2x$ ?

Answer 1

Double angle formulas can be defined as trigonometric formulas that express trigonometric functions of an angle $2x$ in terms of functions of an angle $x$. The formulas are called double angles because trigonometric functions of double angles are involved in them.

Complete step by step solution:
In a right-angled triangle, the trigonometric ratio gives us the relationship between one of the angles of the triangle and the length of its side.
Each trigonometric ratio gives us an idea about any two sides of the right-angled triangle with respect to an angle of the right-angled triangle.
The $\cos 2x$ formula is also one such trigonometric formula and it is also known as the double angle formula because of the double angle $2x$ that is present in the formula.
Since, we are dealing with a double angle, therefore these trigonometric functions are expressed by the addition and/or subtraction of other trigonometric functions or related expressions.
A cosine trigonometric function can be defined as the ratio of the side adjacent to the specified angle to the longest side of the right-angled triangle, which is the hypotenuse.
Now, the three double angle formula for $\cos 2x$ can be given as the following:
$\Rightarrow \cos 2x={{\cos }^{2}}x-{{\sin }^{2}}x$
$\Rightarrow \cos 2x=1-2{{\sin }^{2}}x$
$\Rightarrow \cos 2x=2{{\cos }^{2}}x-1$
The above double angle formulas for the cosine function can be easily deduced or derived from basic trigonometric functions and identities and also some basic algebraic identities as well.
The double angle formulas are also present for the remaining five trigonometric functions as well, which include sine, tangent, cotangent, secant, and cosecant functions.

Note: A trigonometric ratio is derived from the three sides of a right-angled triangle which includes the hypotenuse, the base (side adjacent to the angle specified), and the perpendicular (the side opposite to the angle specified). There are a total of six trigonometric ratios, out of which three are fundamental ratios, which can be used to derive the remaining three functions. The fundamental ratios include the sine, cosine, and tangent functions.