MHT CET · Maths · Trigonometric Ratios & Identities
The value of \((\cos \alpha+\cos \beta)^2+(\sin \alpha+\sin \beta)^2\) is
- A \(2 \sin ^2\left(\frac{\alpha-\beta}{2}\right)\)
- B \(2 \cos ^2\left(\frac{\alpha-\beta}{2}\right)\)
- C \(4 \cos ^2\left(\frac{\alpha-\beta}{2}\right)\)
- D \(4 \sin ^2\left(\frac{\alpha-\beta}{2}\right)\)
Answer & Solution
Correct Answer
(C) \(4 \cos ^2\left(\frac{\alpha-\beta}{2}\right)\)
Step-by-step Solution
Detailed explanation
\( (\cos \alpha+\cos \beta)^2+(\sin \alpha+\sin \beta)^2 \)
\( =\left(\cos ^2 \alpha+\sin ^2 \alpha\right)+\left(\cos ^2 \beta+\sin ^2 \beta\right)+2(\cos \alpha \cdot \cos \beta+\) \(\sin \alpha \cdot \sin \beta) \)
\( =1+1+2 \cos (\alpha-\beta) \)
\( =2\{1+\cos (\alpha-\beta)\} \)
\( =2 \times 2 \cos ^2\left(\frac{\alpha-\beta}{2}\right) \)
\( =4 \cos ^2\left(\frac{\alpha-\beta}{2}\right)\)
\( =\left(\cos ^2 \alpha+\sin ^2 \alpha\right)+\left(\cos ^2 \beta+\sin ^2 \beta\right)+2(\cos \alpha \cdot \cos \beta+\) \(\sin \alpha \cdot \sin \beta) \)
\( =1+1+2 \cos (\alpha-\beta) \)
\( =2\{1+\cos (\alpha-\beta)\} \)
\( =2 \times 2 \cos ^2\left(\frac{\alpha-\beta}{2}\right) \)
\( =4 \cos ^2\left(\frac{\alpha-\beta}{2}\right)\)
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