Trigonometry Addition subtraction formula

This below worksheet help you to understand the Trigonometry Addition and subtraction formulas and their theorem proof.

From Cos(A-B) Theorem Proof
Cos (A-B)= Cos A Cos B + Sin A Sin B ---> 1

To Find Cos(A+B)
Cos(A+B)=Cos(A-(-B))
=Cos A Cos (-B) + Sin A Sin(-B)
=Cos A Cos B +(-)Sin A SinB
=Cos A Cos B - Sin A Sin B --->2

To Find Sin(A-B)
Sin(A-B) = Cos(90-(A-B))
=Cos(90-A+B)
=Cos(B-(A-90))
=Cos B Cos (A-90) + Sin B Sin (A-90)
=Cos B Sin A + Sin B (-)Sin (90 - A)
=Cos B Sin A - Sin B Cos A
=Sin A Con B - Cos A Sin B --->3

To Find Sin(A+B)
Sin(A+B) = Sin ( A - (-B))
apply equation 3 here
Sin (A- (-B))= Sin A Cos(-B) - Cos A Sin (-B)
=Sin A Cos B -Cos A (-)Sin B
=Sin A Cos B + Cos A Sin B --->4

To Find tan(A+B)
tan(A+B)=Sin(A+B)/Cos(A+B)
=(Sin A Cos B + Cos A Sin B)/(Cos A Cos B 1 Sin A Sin B)
=1/(Cos A Cos B) x (Sin A Cos B + Cos A Sin B)/(1-(Sin A/ Cos A) (Sin B/Cos B))
=1/(Cos A Cos B) x (Sin A Cos B + Cos A Sin B)/(1-tan A tan B)
=(Sin A Cos B/Cos A Cos B + Cos A Sin B / Cos A Cos B)/(1-tan A tan B)
=(Sin A/Cos A + Sin B/ CosB)/(1- tan A tan B)
=(tan A + tan B)/(1- tan A tan B) --->5

To Find tan(A-B)
tan(A-B)=Sin(A-B)/Cos(A-B)
=(Sin A Cos B - Cos A Sin B)/(Cos A Cos B + Sin A Sin B)
=1/(Cos A Cos B) x (Sin A Cos B - Cos A Sin B)/(1+(Sin A/ Cos A) (Sin B/Cos B))
=1/(Cos A Cos B) x (Sin A Cos B - Cos A Sin B)/(1+tan A tan B)
=(Sin A Cos B/Cos A Cos B - Cos A Sin B / Cos A Cos B)/(1+tan A tan B)
=(Sin A/Cos A - Sin B/ CosB)/(1+ tan A tan B)
=(tan A - tan B)/(1+ tan A tan B) --->6

To Find Sin(2A)
Sin(2A) = Sin(A+A) --- Apply equation 4
=Sin A Cos A + Cos A SinA
=2SinA CosA --->7

To Find Cos(2A)
Cos(2A)=Cos(A+A) -- Apply equation 2
=Cos A Cos A - Sin A Sin A
=Cos²A - Sin² A --->8

=(1-Sin ²A)- Sin ² A
=1- 2 Sin²A --->9

=Cos²A -( 1 - Cos²A)
=2 Cos²A -1 --->10

Trigonometry Addition and subtraction formulas: