1. Trig addition formulae
sin 𝐴±𝐵 = sin 𝐴 cos 𝐵 ± sin 𝐵 cos 𝐴
Proof
Proof
cos 𝐴±𝐵 = cos 𝐴 cos 𝐵 ∓ sin 𝐵 sin 𝐴
Proof
tan 𝐴±𝐵 = tan 𝐴 ± 𝑡𝑎𝑛 𝐵 1∓ tan 𝐴 tan 𝐵

2. sin (A+B)

3. Geometric proof of 𝐬𝐢𝐧 𝑨+𝑩 = 𝐬𝐢𝐧 𝑨 𝐜𝐨𝐬 𝑩 + 𝐬𝐢𝐧 𝑩 𝐜𝐨𝐬 𝑨
Take a general triangle, as shown below … A B a b h c1 c2 c

4. Now consider the area of the triangle as a whole and as a compound area
Area of red triangle = ½ c1 x h
Area of blue triangle = ½ c2 x h A B a b h c1 c2 c
Area of whole triangle = ½ (c1+c2) x h = ½ ab sin(A+B)

5. Area of whole triangle
= ½ (c1+c2) x h
= ½ ab sin(A+B) A B a b h c1 c2 c
But h = a cos(A) = b cos(B)
with c1 = a sin(A), c2 = b sin(B)

6. Therefore : Area of whole triangle
= ½ (c1+c2) x h = ½ ab sin(A+B)
= ½ (asinA +bsinB)h = ½ ab sin(A+B) A B a b h c1 c2 c
Substituting values of h gives : absinAcosB + absinBcosA = ab sin(A+B)
So finally : sinAcosB + sinBcosA = sin(A+B)

7. cos (A-B)

8. Type equation here.Type equation here.here.
By GCSE Trigonometry: 1 P 1
So the coordinates of P are: B A
Type equation here.Type equation here.here.  O M N
So the coordinates of Q are: Q
𝑆𝑖𝑛𝐴−𝑆𝑖𝑛𝐵 P

9. (𝐶𝑜𝑠𝐴−𝐶𝑜𝑠𝐵 ) 2 +(𝑆𝑖𝑛𝐴−𝑆𝑖𝑛𝐵 ) 2 Multiply out the brackets
𝑃 𝑄 2 =
(𝐶𝑜𝑠𝐴−𝐶𝑜𝑠𝐵 ) 2 +(𝑆𝑖𝑛𝐴−𝑆𝑖𝑛𝐵 ) 2
Multiply out the brackets
𝑃 𝑄 2 =
(𝐶𝑜 𝑠 2 𝐴−2𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵+𝐶𝑜 𝑠 2 𝐵)
+ (𝑆𝑖 𝑛 2 𝐴−2𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵+𝑆𝑖 𝑛 2 𝐵)
Rearrange
𝑃 𝑄 2 =
(𝐶𝑜 𝑠 2 𝐴+𝑆𝑖 𝑛 2 𝐴)
+ (𝐶𝑜 𝑠 2 𝐵+𝑆𝑖 𝑛 2 𝐵)
− 2(𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵+𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵)
𝐶𝑜 𝑠 2 θ+𝑆𝑖 𝑛 2 θ ≡ 1
𝑃 𝑄 2 =
2 − 2(𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵+𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵)

10. You can also work out PQ using the triangle OPQ:Q
You can also work out PQ using the triangle OPQ: Q 1 P 1 1 B P A
B - A 1 O M N
𝑎 2 = 𝑏 2 + 𝑐 2 − 2bcCosA
Sub in the values
𝑃𝑄 2 = − 2Cos(B - A)
Group terms
𝑃𝑄 2 =2 − 2Cos(B - A)
Cos (B – A) = Cos (A – B)
eg) Cos(60) = Cos(-60)
𝑃𝑄 2 =2 − 2Cos(A - B)

11. Cos(A - B) = CosACosB + SinASinB
𝑃 𝑄 2 =
2 − 2(𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵+𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵)
𝑃𝑄 2 =2 − 2Cos(A - B)
2 − 2(𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵+𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵)
= 2 − 2Cos(A - B)
Subtract 2 from both sides
− 2(𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵+𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵)
= − 2Cos(A - B)
Divide by -2
𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵+𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵
= Cos(A - B)
Cos(A - B) = CosACosB + SinASinB
Cos(A + B) = CosACosB - SinASinB

12. tan (A+B)
You may be asked to prove either of the Tan identities using the Sin and Cos ones!

13. Tan (A+B) ≡ 𝑆𝑖𝑛(𝐴+𝐵) 𝐶𝑜𝑠(𝐴+𝐵)
Rewrite
Tan (A+B) ≡ 𝑆𝑖𝑛𝐴𝐶𝑜𝑠𝐵+𝐶𝑜𝑠𝐴𝑆𝑖𝑛𝐵 𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵−𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵
Divide top and bottom by CosACosB
Tan (A+B) ≡ 𝑆𝑖𝑛𝐴𝐶𝑜𝑠𝐵 𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵 + 𝐶𝑜𝑠𝐴𝑆𝑖𝑛𝐵 𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵 𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵 𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵 − 𝑆𝑖𝑛𝐴𝑆𝑖𝑛𝐵 𝐶𝑜𝑠𝐴𝐶𝑜𝑠𝐵
Simplify each Fraction
TanA
+ TanB
Tan (A+B) ≡ 1
- TanATanB