Trigonometric Equations Question 202
Question: In $ \Delta ABC, $ if $ \cos A+\cos C=4{{\sin }^{2}}\frac{1}{2}B, $ then $ a,b,c $ are in
Options:
A) A. P.
B) G. P.
C) H. P.
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
- $ \cos A+\cos C=4{{\sin }^{2}}\frac{1}{2}B $
Þ $ 2\cos \frac{A+C}{2}\cos \frac{A-C}{2}=4{{\sin }^{2}}\frac{B}{2} $
Þ $ \cos \frac{A+C}{2}\cos \frac{A-C}{2}=2{{\sin }^{2}}\frac{B}{2} $
Þ $ \cos ( \frac{A-C}{2} )=2\sin \frac{B}{2} $
Þ $ \cos \frac{A}{2}\cos \frac{C}{2}+\sin \frac{A}{2}\sin \frac{C}{2}=2\sin \frac{B}{2} $
Þ $ \sqrt{\frac{s(s-a)}{bc}}\sqrt{\frac{s(s-c)}{ab}}+\sqrt{\frac{(s-b)(s-c)}{bc}}\sqrt{\frac{(s-a)(s-b)}{ab}} $ $ =2\sqrt{\frac{(s-a)(s-c)}{ac}} $ $ \frac{\sqrt{(s-a),(s-c)}}{ac}+\frac{s-b}{b}\sqrt{\frac{(s-c),\text{ (}s-a\text{)}}{ac}} $ $ =2\sqrt{\frac{(s-a),\text{(}s-c)}{ac}} $
Þ $ \frac{s}{b}+\frac{s-b}{b}=2 $
Þ $ a+c=2b $ Þ $ a,b,c $ are in A. P.
BETA
