Trigonometric Addition and Subtraction formulae

Trigonometric Addition ( Sum ) and Subtraction ( Difference ) formula:

The formulae

which are popularly

known as addition ( sum )

and subtraction( difference )

formulae are as follows:

Sine of sum of angles:

\sin (A+B) = \sin A . \cos B + \cos A . \sin B


Cosine of sum of angles:

\cos (A+B) = \cos A . \cos B - \sin A . \sin B


Sine of difference of angles:

\sin (A-B) = \sin A . \cos B - \cos A . \sin B


Cosine of difference of angles:

\cos (A-B) = \cos A . \cos B + \sin A . \sin B


And similarly the sum and difference of angle formula of Tangent are:

\tan (A+B) = \dfrac{\tan A + \tan B}{1 - \tan A . \tan B}


\tan (A-B) = \dfrac{\tan A - \tan B}{1 + \tan A . \tan B}



Proof of Trigonometric Sum and Difference Formulae:


Now let us prove the identities or formulae listed above.

In the approach to prove these identities we should first prove the identity of \cos (A - B) and rest of the identities can be derived from the identity:

Let P(x_1 , y_1) and Q(x_2 , y_2) be two points , different from the origin , on the terminal arms of two angles A and B such that A > B placed in the standard position as shown in the figure below:

cosine difference formula

cosine difference formula



If: OP = r_1 and OQ = r_2 then,

x_1 = r_1 . \cos A , y_1 = r_1 . \sin A


and x_2 = r_2 . \cos B , y_2 = r_2 . \sin B


Hence , Using the Distance formula:

PQ^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2


or , PQ^2 = (x^2_1 + y^2_1) + (x^2_2 + y^2_2) - 2(x_1 x_2 + y_1 y_2


or, PQ^2 = r^2_1 + r^2_2 - 2 r_1 r_2 (\cos A . \cos B + \sin A \sin B)   ————–  Expression 1 for PQ^2


Now , Let us rotate the coordinate system so that OQ coincides with the positive x-axis. This is equivalent to place the angle (A-B) in the standard position with OQ along the positive x-axis. As shown in figure below:

cosine difference formula

cosine difference formula


Then , the new coordinates of P(x^| , y^|) and Q(x^{||} , y^{||}) are given by:

x^| = r_1 \cos (A-B)


, y^| = r_1 \sin (A-B)


and :

x^{||} = r_2 and y^{||} = 0


Hence now:

PQ^2 = (x^{||} - x^|)^2 + (y^{||} - y^|)^2


Or , PQ^2 = x^{|2} + y^{|2} + x^{||2} - 2 x^| x^{||}


Or , PQ^2 = r^2_1 + r^2_2 - 2 r_1 r_2 . \cos (A - B)   ————–  Expression 2 for PQ^2


Now comparing the two expression for PQ^2 , we have:

2 r_1 r_2 . \cos (A - B) = 2 r_1 r_2 (\cos A . \cos B + \sin A \sin B)


Hence: \cos (A - B) = \cos A . \cos B + \sin A . \sin B

This is the subtraction or difference formula for cosine , true for arbitrary angles A and B.



Now we can derive other sum and difference formula using this formula, before deriving those formulae let us consider following special cases:


Special Cases:

Note: All the numbers and variables eg: A , B , 90 are in Degrees , but we can also apply this in other angle measurement system by replacing the value of numbers used with corresponding value.

Case 1> If A=o then , we get:

\cos(-B) = \cos B


Case 2> If A=90 then, we get:

\cos(90 - B) = \sin B


Case 3> If B is replaced by 90-B  in Case 2 then we get:

\cos B = \sin (90 - B)


Case 4> If B is replaced by -90 and A is replaced by -A then , we get:

- \sin (-A) = \sin A and \sin (-A) = - \sin A


Now let us Derive other general formulas using the special cases above and cosine difference of angle formula:


Formula for \cos (A+B)

Replacing B by -B in \cos (A - B) = \cos A . \cos B + \sin A . \sin B we get:

\cos (A + B) = \cos A . \cos (-B) + \sin A . \sin (-B)

Now using case 2 and 5 we get:

\cos (A + B) = \cos A . \cos B - \sin A . \sin B


Formula for \sin (A+B)

Replacing A by 90-A to get:

\cos (90-A-B) = \cos (90-A) . \cos B + \sin (90 - A) . \sin B

Then , using Case 3 and 4 we get:

\sin (A+B) =\sin A \cos B + \cos A \sin B


Formula for \sin (A-B)

Replacing  B by -B in above formula or  \sin (A+B) =\sin A \cos B + \cos A \sin B   and the uisng case 2 and 5 we get:

\sin (A-B) =\sin A \cos B - \cos A \sin B

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