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

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:

*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:

*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 terms of cos:

*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

*then we get*

**cos2x formula***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*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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