# DOUBLE ANGLE FORMULA IN TRIGONOMETRY

## Double Angle Formulae

Use an appropriate 'double angle' formula to simplify:

Problem 1 :

2 sin α cos α

Solution:

= 2 sin α cos α

= sin 2α

Problem 2 :

4 cos α sin α

Solution:

= 4 cos α sin α

= 2 × 2 cos α sin α

= 2 sin 2α

Problem 3 :

sin α cos α

Solution:

= sin α cos α

Multiplying and dividing by 2, we get

= (2/2) sin α cos α

= (1/2) (2sin α cos α)

= (sin 2α/2)

Problem 4 :

2 cos2 β - 1

Solution:

= 2 cos2 β - 1

= cos 2β

Problem 5 :

1 - 2 cos2 ϕ

Solution:

= 1 - 2 cos2 ϕ

= - (2 cos2 ϕ - 1)

= - cos 2ϕ

Problem 6 :

1 - 2 sin2 N

Solution:

= 1 - 2 sin2 N

= cos 2N

Problem 7 :

2 sin2 M - 1

Solution:

= 2 sin2 M - 1

= - (1 - 2 sin2 M)

= - cos 2M

Problem 8:

cos2 α - sin2 α

Solution:

= cos2 α - sin2 α

= cos 2α

Problem 9 :

sinα - cos2 α

Solution:

= sin2 α - cos2 α

= - (cos2 α - sin2 α)

= - cos 2α

Problem 10 :

2 sin 2A cos 2A

Solution:

= 2 sin 2A cos 2A

= sin (2 × 2A)

= sin 4A

Problem 11 :

2 cos 3α sin 3α

Solution:

= 2 cos 3α sin 3α

= sin (2 × 3α)

= sin 6α

Problem 12 :

2 cos2 4θ - 1

Solution:

= 2 cos2 4θ - 1

It is in the form of 2 cos2 θ - 1, instead of θ we have 4θ

= cos (2 × 4θ)

= cos 8θ

Problem 13 :

1 - 2 cos2

Solution :

= 1 - 2 cos2

= - (2 cos2 3β - 1)

= - cos (2 × 3β)

= - cos 6β

Problem 14 :

1 - 2 sin2 5α

Solution:

= 1 - 2 sin2

= cos (2 × 5α)

= cos 10α

Problem 15 :

2 sin2 3D - 1

Solution:

= 2 sin2 3D - 1

= - (1 - 2 sin2 3D)

= - cos (2 × 3D)

= - cos 6D

Problem 16 :

cos2 2A - sin2 2A

Solution:

= cos2 2A - sin2 2A

The given question looks like cos2 A - sin2 A, Here A = 2A

cos2 A - sin2 A = cos 2A

cos2 2A - sin2 2A = cos 2(2A)

= cos 4A

Problem 17 :

cos2 (α/2) - sin2 (α/2)

Solution:

= cos2 (α/2) - sin2 (α/2)

The given question looks like cos2 A - sin2 A, Here A = α/2

cos2 A - sin2 A = cos 2A

= cos (2 × α/2)

= cos α

Problem 18 :

2 sin2 3P - 2 cos2 3P

Solution:

= 2 sin2 3P - 2 cos2 3P

Factoring 2

= 2(sin2 3P - cos2 3P)

= -2(cos2 3P - sin2 3P)

= 2(cos(2 × 3P))

= - 2 cos (2 × 3P)

= -2 cos 6P

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