What is the Formula of (1) 1+cos 2x and (2) 1-cos2x

What is the formula of (1) 1+ cos 2x and (2) 1-cos2x?

Answer:

The branch of mathematics that deals with the relationships between the lengths and angles of a triangle is known as Trigonometry.

Trigonometric identities are mathematical expressions that are made up of functions. These are various identities that are derived from basic functions that are sin, cos, tan, etc. 

 (1) 1 + cos2x = 2cos2 x,  

 (2) 1 – cos2x = 2sin2x

Explanation:

Using the trigonometric formula, 

Cos (a + b) formula is

Cos (a + b) = Cos a. cos b – Sin a. Sin b                              (1)

but when a is the same as b then

Cos 2a = Cos a. Cos a – Sin a. Sin a

Cos 2a = Cos2a – Sin2a                                                           (2)

As per the formula, we have

Cos2a + Sin2a = 1                                                                     (3)

Let us consider the first given equation 1 + Cos 2x

1 + Cos 2x

From equation (2) we can write it as

1 + Cos 2x = 1 + Cos2x – Sin2x                                                 (4)

From equation (3) 1 is equal to Cos2x + Sin2x.

 Substitute this formula in equation (4)

1 + Cos 2x = Cos2x + Sin2x + Cos2x – Sin2x

1 + Cos 2x = 2Cos2x

Similarly for the second given equation 

1 – Cos 2x = 1 – Cos2x – Sin2x                                 (5)

From equation (3) 1 is equal to Cos2x + Sin2x.

Now we can substitute this value in the equation (5)

1 – Cos 2x = Cos2x + Sin2x – Cos2x Sin2x

1 – Cos 2x = 2 Sin2x