Cos2x formula in maths explained

Learn and know what is cos2x formula in trigonometry chapter. This is one of the important formulas in multiple angles concept.

Cos2x formula in maths explained

Totally there are 4 different ways we have to express cos2x formula. There is a general formula for cos2x other than this we also write cos2x in terms of sin, cos2x in terms of cos and finally cos2x in terms of tan. All these formulas how we got i.e. derivations are given below, go through step by step you can understand it without any difficulty.

Cos2x formula proof (derivation) as follows:

We know that, Cos(A+B) = CosA x CosB – SinA x SinB. This formula we will make use and derive the formula for cos2x.

In the above cos(A+B) formula, we will replace A and B with “x” and find cos2x value.

Cos(x+x) = Cosx × Cosx – Sinx × Sinx

Cos2x = \cos ^{ 2 }{ x } \sin ^{ 2 }{ x }

Cos2x formula in terms of sin:

Now we will learn how to write cos2x in terms of sin. For this we should have idea about trigonometric identities.

We know that, \sin ^{ 2 }{ x } + \cos ^{ 2 }{ x } = 1. From this identity, we will find the value of \cos ^{ 2 }{ x } . Therefore, \cos ^{ 2 }{ x } = 1 – \sin ^{ 2 }{ x } . We will replace this \cos ^{ 2 }{ x } value in cos2x then we will get cos2x formula in terms of sin.

Cos2x = \cos ^{ 2 }{ x } \sin ^{ 2 }{ x }

            = 1 – \sin ^{ 2 }{ x } \sin ^{ 2 }{ x }

            = 1 – 2 \sin ^{ 2 }{ x }

Finally the cos2x in terms of sin is equal to 1 – 2 \sin ^{ 2 }{ x }

Cos2x formula in maths explained

Cos2x formula in terms of cos:

How to write cos2x formula in terms of cos, we will learn now. Already we know that \sin ^{ 2 }{ x } + \cos ^{ 2 }{ x } = 1. From this find out \sin ^{ 2 }{ x } value. We get \sin ^{ 2 }{ x } = 1 – \cos ^{ 2 }{ x } . We will \sin ^{ 2 }{ x } value in cos2x formula then we get cos2x in terms of cos only.

Cos2x = \cos ^{ 2 }{ x } \sin ^{ 2 }{ x }

           = \cos ^{ 2 }{ x } – (1- \cos ^{ 2 }{ x } )

            = \cos ^{ 2 }{ x } – 1 + \cos ^{ 2 }{ x }

           = 2 \cos ^{ 2 }{ x } -1

Finally the cos2x in terms of cos is equal to 2 \cos ^{ 2 }{ x } -1

Cos2x formula in terms of tan:

Cos2x     = \cos ^{ 2 }{ x } \sin ^{ 2 }{ x }

                = \cos ^{ 2 }{ x } (1- \frac { \sin ^{ 2 }{ x } }{ \cos ^{ 2 }{ x } } ) (take \cos ^{ 2 }{ x } as common)

                = \frac { 1 }{ \sec ^{ 2 }{ x } } (1-\tan ^{ 2 }{ x } )

                = \frac { 1-\tan ^{ 2 }{ x } }{ 1+\tan ^{ 2 }{ x } }

Therefore, cos2x value in terms of tan is equal to \frac { 1-\tan ^{ 2 }{ x } }{ 1+\tan ^{ 2 }{ x } } .

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