## Double Angle Formulas

In this page we are going to see double angle formulas and also we are going to see the example problems.

1.Sin 2A = 2 Sin A cos A

2.Cos 2A = cos² A - Sin² A

3. tan 2A = 2 tan A/(1-tan² A)

4.Cos 2A = 1 - 2Sin² A

5.Cos 2A = 2Cos² A - 1

6. sin 2A = 2 tan A/(1+tan² A)

7.cos 2A = (1-tan² A)/(1+tan² A)

8. sin²A = (1-Cos 2A)/2

9.Cos²A = (1+Cos 2A)/2

Now we are going to see the example problems based on the above formulas.

Example 1:

Prove that 2 Sin 15° cos 15° = 1/2

L.H.S

2 Sin 15° cos 15°

This looks like the formula 2 Sin A cos A.Now we have to apply the formula.The required formula is

Sin 2A = 2 Sin A cos A

Instead of A we have 15°

2 Sin 15° cos 15° =  Sin 2(15°)

=  Sin 30°

=  1/2

R.H.S

Hence proved             double angle formulas

Example 2:

Show that Cos 20° cos 40° cos 80° = 1/8

L.H.S

Cos 20° cos 40° cos 80°

=Cos 20° cos (60° - 20°) cos (60° + 20°)

=Cos 20° cos (60° - 20°) cos (60° + 20°)

Cos (A-B) Cos (A+B) = Cos² A - Sin² A

= Cos 20°[cos ² 60°- Sin² 20°]

= Cos 20°[(1/2)² - Sin² 20°]

= Cos 20°[1/4 -Sin² 20° ]

= Cos 20°(1 -4Sin² 20°)/4

= Cos 20°(1 -4(1-cos² 20°)/4)

= Cos 20°(1 -4 + 4cos² 20°)/4

= Cos 20°(-3 + 4cos² 20°)/4

= (-3Cos 20° + 4cos³ 20°)/4

= ( 4cos³ 20° - 3Cos 20°)/4

= Cos (3 x 20°)/4

= Cos (3 x 20°)/4

=  Cos 60°/4

=  (1/2) /4

=  1/8

R.H.S

Hence proved

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