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:

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

and,

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

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

Now,

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

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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