Question

In a $\Delta ABC$; the value of $\frac{co{s}^2\frac{B-C}{2}}{(b+c{)}^2}+\frac{si{n}^2\frac{B-C}{2}}{(b-c{)}^2}$ is

• A. $a^2$
• B. $\frac{1}{a^2}$
• C. $\frac{b^2+c^2}{a^2}$
• D. $\frac{b^2-c^2}{a^2}$

Answer 1

The correct option is B $\frac{1}{a^2}$

$\frac{co{s}^2\frac{B-C}{2}}{4R^2.4si{n}^2\frac{B+C}{2}.co{s}^2\frac{B-C}{2}}+\frac{si{n}^2\frac{B-C}{2}}{4R^2.4co{s}^2\frac{B+C}{2}.si{n}^2\frac{B-C}{2}}=\frac{1}{16R^2}[\frac{1}{co{s}^2\frac{A}{2}}+\frac{1}{si{n}^2\frac{A}{2}}]=\frac{1}{4.R.{(2sin\frac{A}{2}.cos\frac{A}{2})}^2}=\frac{1}{4R^{2}si{n}^{2}A}=\frac{1}{a^2}$