2x 3x formula - Proving

Chapter 3 Class 11 Trigonometric Functions
Concept wise

These formulas can be derived using x + y formulas

### For sin 2x

sin 2x = sin (x + x)

Using sin (x + y) = sin x cos y + cos x sin y

= sin x cos x + sin x cos x

= 2 sin x cos x

### For cos 2x

cos 2x = cos (x + x)

Using cos (x + y) = cos x cos y – sin x sin y

= cos x cos x – sin x sin x

= cos 2 x – sin 2 x

Putting cos 2 x = 1 – sin 2 x

cos 2x = cos 2 x – sin 2 x

= 1 – sin 2 x – sin 2 x

= 1 – 2sin 2 x

Putting sin 2 x = 1 – cos 2 x

cos 2x = cos 2 x – sin 2 x

= cos 2 x – (1 – cos 2 x)

= cos 2 x – 1 + cos 2 x

= 2cos 2 x – 1

### Transcript

These formulas can be derived using x + y formulas For sin 2x sin 2x = sin (x + x) Using sin (x + y) = sin x cos y + cos x sin y = sin x cos x + sin x cos x = 2 sin x cos x For cos 2x cos 2x = cos (x + x) Using cos (x + y) = cos x cos y – sin x sin y = cos x cos x – sin x sin x = cos2 x – sin2 x Putting cos2 x = 1 – sin2 x cos 2x = cos2 x – sin2 x = 1 – sin2 x – sin2 x = 1 – 2sin2 x Putting sin2 x = 1 – cos2 x cos 2x = cos2 x – sin2 x = cos2 x – (1 – cos2 x) = cos2 x – 1 + cos2 x = 2cos2 x – 1 For tan 2x