Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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If Cos α + Cos β = 1 3 and Sin Sin α + Sin β = 1 4 , Prove that Cos α − β 2 = ± 5 24 - Mathematics

Numerical

If \[\cos \alpha + \cos \beta = \frac{1}{3}\]  and sin \[\sin\alpha + \sin \beta = \frac{1}{4}\] , prove that \[\cos\frac{\alpha - \beta}{2} = \pm \frac{5}{24}\]

 
 

 

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Solution

Squaring and adding equations

\[\cos \alpha + \cos \beta = \frac{1}{3}\]  and \[\sin\alpha + \sin \beta = \frac{1}{4}\] , we get
\[\left( \cos^2 \alpha + \cos^2 \beta + 2cos\alpha \times cos\beta \right) + \left( \sin^2 \alpha + \sin^2 \beta + 2sin\alpha \times sin\beta \right) = \frac{1}{9} + \frac{1}{16}\]
\[ \Rightarrow 1 + 1 + 2\left( cos\alpha \times cos\beta + sin\alpha \times sin\beta \right) = \frac{25}{144}\]
\[ \Rightarrow 2 + 2\cos\left( \alpha - \beta \right) = \frac{25}{144} \left( \because \cos\left( A - B \right) = \text{ cos } A \times \text{ cos }B + \text{ sin } A \times \text{ sin } B \right)\]
\[ \Rightarrow \cos\left( \alpha - \beta \right) = - \frac{263}{288} . . . (1)\]
Now,
\[\cos^2 \left( \frac{\alpha - \beta}{2} \right) = \frac{1 + \cos\left( \alpha - \beta \right)}{2}\]
\[ = \frac{1 - \frac{263}{288}}{2} [\text{ From }  (1)]\]
\[ = \frac{25}{576}\]
\[ = \pm \frac{5}{24}\]
 
 
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 42 | Page 30
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