Double angle formulas.jpg

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

 

For tan 2x

for tan 2x.jpg

  1. Chapter 3 Class 11 Trigonometric Functions
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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

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