# If Sin α + Sin β = a and Cos α + Cos β = B , Prove that (Ii) Cos ( α − β ) = a 2 + B 2 − 2 2 - Mathematics

Numerical

If $\sin \alpha + \sin \beta = a \text{ and } \cos \alpha + \cos \beta = b$ , prove that

(ii) $\cos \left( \alpha - \beta \right) = \frac{a^2 + b^2 - 2}{2}$

#### Solution

The given equations are $\sin \alpha + \sin \beta = a \text{ and } \cos \alpha + \cos \beta = b$

$\text{ On squaring } sin\alpha + sin\beta = \text{ a and } cos\alpha + cos\beta = \text{ b and adding them, we get}$
$\sin^2 \alpha + \sin^2 \beta + 2 \times sin\alpha sin\beta + \cos^2 \alpha + \cos^2 \beta + 2 \times cos\alpha cos\beta = a^2 + b^2$
$\Rightarrow 1 + 1 + 2\left( sin\alpha sin\beta + cos\alpha cos\beta \right) = a^2 + b^2$
$\Rightarrow 2\left( sin\alpha sin\beta + cos\alpha cos\beta \right) = a^2 + b^2 - 2$
$\Rightarrow 2\cos\left( \alpha - \beta \right) = a^2 + b^2 - 2 \left( \because \cos\left( A - B \right) = sinAsinB + cosAcosB \right)$
$\Rightarrow \cos\left( \alpha - \beta \right) = \frac{a^2 + b^2 - 2}{2}$

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.1 | Q 38.2 | Page 29