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Trigonometric Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsTrigonometric Functions2 Marks
MCQ
In $\triangle A B C$, if $\sin ^2 \frac{A}{2}, \sin ^2 \frac{B}{2}, \sin ^2 \frac{C}{2}$ be in H.P., then a, b, c will be in
  • A
    A. P.
  • B
    G. P.
  • ✓
    H. P.
  • D
    A.G.P.
✓

Answer

Correct option: C.
H. P.
(C) $\frac{1}{\sin ^2 \frac{\Lambda}{2}}, \frac{1}{\sin ^2 \frac{B}{2}}, \frac{1}{\sin ^2 \frac{C}{2}}$ are in A.P.
$\Rightarrow \frac{1}{\sin ^2 \frac{C}{2}}-\frac{1}{\sin ^2 \frac{B}{2}}=\frac{1}{\sin ^2 \frac{B}{2}}-\frac{1}{\sin ^2 \frac{A}{2}}$
$\Rightarrow \frac{a b}{(s-a)(s-b)}-\frac{a c}{(s-a)(s-c)}$ $=\frac{a c}{(s-a)(s-c)}-\frac{b c}{(s-b)(s-c)}$
$\Rightarrow\left(\frac{ a }{ s - a }\right)\left(\frac{ b ( s - c )- c ( s - b )}{( s - b )( s - c )}\right)$ $=\left(\frac{ c }{ s - c }\right)\left(\frac{ a ( s - b )- b ( s - a )}{( s - a )( s - b )}\right)$
$\begin{array}{l}\Rightarrow a b s-a b c-a c s+a b c=a c s-a b c-b c s+a b c \\ \Rightarrow a b-a c=a c-b c \Rightarrow a b+b c=2 a c\end{array}$
$\Rightarrow \frac{1}{ c }+\frac{1}{ a }=\frac{2}{b} \Rightarrow a , b , c$ are in H.P.

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