The double angle identities: sin2A, cos2A and tan 2A derived from the trigonometric addition formulas.

The double angle trigonometric identities can be derived from the addition trigonometric identities:

Basically, all you need to do change all of the B’s to A’s.

Let’s start off with the sine addition identity:

sin (A+B) = sinAcosB + cosAsinB

Now changing the B’s to A’s you get:

sin(A+A) = sinAcosA + cosAsinA

This simplifies down to:

sin(2A) = 2sinAcosA

Next, let’s derive the cosine double angle trigonometric identity. There are three different versions of this!

First start off with the cosine addition identity:

cos (A+B) = cosAcosB – sinAsinB

As with sine, all you need to do now is replace the B’s to A’s:

This gives:

cos (A+A) = cosAcosA – sinAsinA

cos(2A) = cos²A - sin²A

Now you can use the well known identity, cos²A + sin²A = 1, to change the cos²A and sin²A to give two further identities:

First, replace cos²A with 1 - sin²A in cos(2A) = cos²A - sin²A:

cos (2A) = 1 - sin²A - sin²A

cos(2A) = 1 – 2sin²A

To get the final identity, this time substitute sin²A = 1 - cos²A into cos(2A) = cos²A - sin²A:

cos(2A) = cos²A – (1-cos²A)

cos(2A) = cos²A – 1 + cos²A

cos(2A) = 2cos²A – 1

So cos(2A) = cos²A - sin²A = 1 – 2sin²A = 2cos²A – 1

Finally, all we need to do now is derive the tangent double angle formula.

Like the previous examples, change the B’s to A’s in the addition formula for tangent:

tan (A+B) = (tanA + tanB)÷ (1 – tanAtanB)

tan (A+A) = (tanA + tanA)÷ (1 – tanAtanA)

tan(2A) = (2tanA) ÷ (1 - tan²A)

So there you have the 3 double angle trigonometric identities:

sin(2A) = 2sinAcosA

cos(2A) = cos²A - sin²A = 1 – 2sin²A = 2cos²A – 1

tan(2A) = (2tanA) ÷ (1 - tan²A)

Just remember there are 3 version of the double angle cosine formula.

For more help on the trigonometric addition and difference formula click here.

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