The word trigonometry being driven from the Greek Word “trigon, and “metron, and it means measuring the sides of triangle The subject originally thought and part of the scope of development to solve geometric problems involving triangles. We know about the trigonometric ratio of the acute angles as the ratio of the sides of the right sides of the right-angle triangle. In this article, we will discuss the concept and COS2X formula of one such trigonometric ratios namely cos2x with other trigonometric ratios.

**COS 2X FORMULA**

**WHAT IS COS2X?**

The trignometric ratio of an angle in a right triangle define the the reltionship between the angle and the length of its sides .Cosin2x or Cos2X is also one such trignometrical formula also known as double angle formula as it as double angle on it.Because of this ,it is being driven by the expression for trignometric function of the sum and difference of two numbers and related expressions.

**DERIVATION OF COS2X FORMULA?**

Let us start with the consideration of addition formula

**COS (X+Y)**

**cos X cos Y -sin X sin Y**** **

Let us equate ,X and Y ,I.e. X=Y

So the above formula for cos2x becomes

**Cos 2X=cos(X+X)**

**CosX CosX -sin X sinX**

**Cos 2X =cos****2****X**** -sin****2 x**

Hence the first cos2x follow as

**Cos2X = cos****2****X**** -sin****2 x**

And for this reason ,we know this formula as double the angle formula ,because we double the angle

**OTHER FORMULAS OF COS2X**

**COS2X =1-2sin****2 x**

To derive this we need to start from the earlier derivation As we already know that

cos2X =Cos2X -sin2X

Cos2X =(1-sin2 X ) -sin2 X (Since ,cos2X=(1-sin2 X )

cos2X=1-sin2 X -sin2 X

So

Cos2X=1-(sin2 X+sin2 X)

Hence cos2x =1-2sin2 X

Cos2x =2COS2X-1

To derive this we need to start from the eariler derivation As we already know that

cos2X =Cos2X -sin2X

Cos2x=cos2X-(1-cos2X){Since sin2x=(1-cos2X)}

Cos2x =cos2X-1 +cos2X

cos2X=(cos2X+cos2X)-1

Hence cos2X=2cos2X-1

cos2X=1-tan2X\1+tan2X

To derive this we need to start from the earlier derivation As we already know that

cos2X=cos2X-sin2X

cos2X=cos2X-sin2X

cos2X=cos2X-sin2X\1

cos2X=cos2X-cos2X\cos2X+cos2X{since cos2X+sin2X=1}

Dividind both numerator and demonetor by cos2X we get

Cos 2X=1-tan2X\1+tan2X{Since that tanx=sinx\cosx)

**HENCE COS2X =1-tan2X\1+tan2X**

**SOLVED EXAMPLE**

Now in order to understand this formula completeley there is an example

**QURSTION PROVE THAT **

**COS3X=4COS****3**** X-3COS X**

**SOLUTION**

COS3X=cos (2x+x)

COS3X=cos2x cosx-sin2x sinX

COS3X=cos2xcosx-sin2x sinx

COS3X=(2cos2x-1)cosX-2sinXcos X sinX

COS3X=(2COS2x-1)cosX -2cosx (1-COS2X)

COS3X=2COS3X-cosX -2cosX+2COS3X

COS3X**=**4COS3 X-3COS X

Hence proved

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In Progress
04-06-2020
Subhani majoka
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