2x 3x formula - Proving

Chapter 3 Class 11 Trigonometric Functions
Concept wise

These formulas can be derived using x + y formulas

### For sin 3x

sin 3x = sin (2x + x)

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

= sin 2x cos x + sin x cos 2x

Using cos 2x = 1 – 2 sin 2 x and sin 2x = 2 sin x cos x

= (2 sin x cos x) cos x + sin x (1 – 2sin 2 x)

= 2 sin x cos 2 x + sin x – 2sin 3 x

Using cos 2 x = 1 – sin 2 x

= 2 sin x (1 – sin 2 x) + sin x – 2sin 3 x)

= 2 sin x – 2sin 3 x + sin x – 2sin 3 x

= 3 sin x – 4 sin 3 x

### For cos 3 x

cos 3x = cos (2x + x)

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

= cos 2x cos x – sin 2x sin x

Using cos 2x = 2 cos 2 x – 1 and sin 2x = 2 sin x cos x

= (2cos 2 x – 1) cos x – (2 sin x cos x) sin x

= 2cos 2 x – cos x – 2 sin 2 x cos x

Using sin 2 x = 1 – cos 2 x

= 2cos 3 x – cos x – 2 (1 – cos 2 x) cos x

= 2cos 3 x – cos x – 2 cos x (1 – cos 2 x)

= 2cos 3 x – cos x – 2 cos x + 2 cos 3 x

= 4cos 3 x – 3cos x

### Transcript

These formulas can be derived using x + y formulas For sin 3x sin 3x = sin (2x + x) Using sin (x + y) = sin x cos y + cos x sin y = sin 2x cos x + sin x cos 2x Using cos 2x = 1 – 2 sin2 x and sin 2x = 2 sin x cos x = (2 sin x cos x) cos x + sin x (1 – 2sin2 x) = 2 sin x cos2 x + sin x – 2sin3 x Using cos2 x = 1 – sin2 x = 2 sin x (1 – sin2 x) + sin x – 2sin3 x) = 2 sin x – 2sin3 x + sin x – 2sin3 x = 3 sin x – 4 sin3 x For cos 3x cos 3x = cos (2x + x) Using cos (x + y) = cos x cos y – sin x sin y = cos 2x cos x – sin 2x sin x Using cos 2x = 2 cos2 x – 1 and sin 2x = 2 sin x cos x = (2cos2 x – 1) cos x – (2 sin x cos x) sin x = 2cos2 x – cos x – 2 sin2 x cos x Using sin2 x = 1 – cos2 x = 2cos2 x – cos x – 2 (1 – cos2 x) cos x = 4cos2 x – 3cos x For tan 3x