AP EAMCET · Maths · Trigonometric Ratios & Identities
If \(\alpha, \beta, \gamma\) are any three angles, then \(\cos \alpha+\cos \beta-\cos \gamma-\cos (\alpha+\beta+\gamma)=\)
- A \(4 \cos \frac{\alpha+\beta}{2} \cos \frac{\beta+\gamma}{2} \cos \frac{\gamma+\alpha}{2}\)
- B \(4 \cos \frac{\alpha+\beta}{2} \sin \frac{\beta+\gamma}{2} \sin \frac{\gamma+\alpha}{2}\)
- C \(4 \cos \frac{\alpha+\beta}{2} \sin \frac{\beta-\gamma}{2} \sin \frac{\gamma-\alpha}{2}\)
- D \(4 \sin \frac{\alpha+\beta}{2} \cos \frac{\beta+\gamma}{2} \cos \frac{\gamma+\alpha}{2}\)
Answer & Solution
Correct Answer
(B) \(4 \cos \frac{\alpha+\beta}{2} \sin \frac{\beta+\gamma}{2} \sin \frac{\gamma+\alpha}{2}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \cos \alpha+\cos \beta-\cos \gamma-\cos (\alpha+\beta+\gamma) \\ & =2 \sin \frac{2 \alpha+\beta+\gamma}{2} \sin \frac{\beta+\gamma}{2}+2 \sin \frac{\beta+\gamma}{2} \sin \left(\frac{\gamma-\beta}{2}\right) \\ & =2 \sin \frac{\beta+\gamma}{2}\left[\sin \frac{2…
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