Multiple Angle Identities

Derive the double angle identity for sin: sin 2A = 2 sin A cos A

sin 2A = sin (A + A)

= sin A cos A + sin A cos A

= 2 sin A cos A

So sin 2A = 2 sin A cos A

Derive the double angle identities for cos: cos 2A = cos²A - sin²A

cos 2A = 1 - 2 sin²A
cos 2A = 2 cos²A - 1

cos 2A = cos (A + A)

= cos A cos A - sin A sin A

= cos²A - sin²A

So cos 2A = cos²A - sin²A
cos 2A = cos²A - sin²A

= (1 - sin²A) - sin²A

cos²A + sin²A = 1

= 1 - 2 sin²A

So cos 2A = 1 - 2 sin²A
cos 2A = cos²A - sin²A

= cos²A - (1 - cos²A)

cos²A + sin²A = 1

= 2 cos²A - 1

So cos 2A = 2 cos²A - 1

Derive the double angle identity for tangent: tan 2A = 2 tan A

1 - tan²A

tan 2A = sin 2A

cos 2A

= 2 sin A cos A (Double angle identities for sine and cosine)
cos²A - sin²A

                  2 sin A
            =        cos A         (Divide the numerator and denominator by cos²A)
                  1 - sin²A
                       cos²A

= 2 tan A
1 - tan²A

So tan 2A = 2 tan A

1 - tan²A

Derive the half angle identity for cosine:

cos 2B = 2 cos²B - 1 (starting with a double angle identity for cosine)
cos 2B + 1 = 2 cos²B (adding 1 to both sides)
1 + cos 2B = cos²B (divide both sides by 2)

This statement is known as a power-reducing formula

(take the square root of both sides)

Now, let B = A and 2B = A, and substitute it into the last expression.

Derive the half angle identity for sine:

cos 2B = 1 - 2 sin²B (starting with a double angle identity for cosine)
cos 2B - 1 = - 2 sin²B (subtracting 1 from both sides)
1 - cos 2B = sin²B (divide both sides by -2)

This statement is known as a power-reducing formula

(take the square root of both sides)

Now, let B = A and 2B = A, and substitute it into the last expression.

Derive the half angle identities for tangent:

                                                            tan A = \/(1 - cos A)
                                                                  2       \/(1 + cos A)
                                                             tan A = 1 - cos A
                                                                    2          sin A
                                                              tan A =      sin A
                                                                    2       1 + cos A

tan A = sin (A/2) (tangent identity)

2 cos (A/2)

              \/(1 - cos A)
           =          \/2           (use half angle identities for sine and cosine)
              \/(1 + cos A)
                       \/2

= \/(1 - cos A) (invert and multiply)
\/(1 + cos A)

So tan A = \/(1 - cos A)

2 \/(1 + cos A)

tan²B = 1 - cos B
1 + cos B is known as a power-reducing formula

tan A = \/(1 - cos A) (the above half angle identity for tangent)

2 \/(1 + cos A)

= \/(1 - cos²A) (rationalize the denominator)
1 + cos A

= \/sin²A (use the Pythagorean Identity)
1 + cos A

= sin A (simplify the square root)
1 + cos A

So tan A = sin A

2 1 + cos A

tan A = \/(1 - cos A) (the first half angle identity for tangent)

2 \/(1 + cos A)

= 1 - cos A (rationalize the numerator)
\/(1 - cos²A)

= 1 - cos A (use the Pythagorean Identity)
\/sin2A

= 1 - cos A (simplify the square root)
sin A

So tan A = 1 - cos A

2 sin A

Product to Sum Formulas:

sin u sin v = (1/2)[cos (u - v) - cos (u + v)]

(1/2)[cos (u - v) - cos (u + v)]

= (1/2)[cos u cos v + sin u sin v - (cos u cos v - sin u sin v)]

= (1/2)[cos u cos v + sin u sin v - cos u cos v + sin u sin v]

= (1/2)(2 sin u sin v)

= sin u sin v

The next three are proven in a similar fashion

cos u cos v = (1/2)[cos (u - v) + cos (u + v)]
sin u cos v = (1/2)[sin (u + v) + sin (u - v)]
cos u sin v = (1/2)[sin (u + v) - sin (u - v)]

Sum to Product Formulas

sin x + sin y = 2 sin x + y cos x - y

2 2

Let u = x + y and v = x - y

2 2

Then 2u = x + y and

2v = x - y

Adding, 2(u + v) = 2x so u + v = x

2(u - v) = 2y

u - v = y

Substituting,

sin x = sin (u + v)

sin y = sin (u - v)

From the product to sum formula: sin u cos v = (1/2)[sin (u + v) + sin (u - v)]

2 sin u cos v = sin (u + v) + sin (u - v)

The next three formulas are proven in a similar fashion

sin x - sin y = 2 cos x + y sin x - y

2 2

cos x + cos y = 2 cos x + y cos x - y

2 2

cos x - cos y = - 2 sin x + y sin x - y

2 2

Problems

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