Prove that cos 2 1 2 B-C b-c 2 -sin 2 1 2 B-C b-c 2 - 1 a 2

L.H.S=cos ^ 2 ( B C 2 ) / k ^ 2 ( sin B +sin C ) ^ 2 +sin ^2 ( B C 2 ) / k ^ 2 ( sin B sin C ) ^ 2 =1/4 k ^ 2 sin ^ 2 ( B+C 2 ) +1/4 k ^ 2 ...

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Integration of Trigonometric Functions

Prove that ...

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Prove that ${cos}^{2} \frac{{\frac{1}{2} (B - C)}}{{(b + c)}^{2}} + {sin}^{2} \frac{{\frac{1}{2} (B - C)}}{{(b - c)}^{2}} = \frac{1}{a^{2}}$

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c^{2} = {(a - b)}^{2} {cos}^{2} \frac{1}{2} C + {(a + b)}^{2} {sin}^{2} \frac{1}{2} C

\frac{{cos}^{2} \frac{B - C}{2}}{(b + c)^{2}} + \frac{{sin}^{2} \frac{B - C}{2}}{(b - c)^{2}} = \frac{1}{a^{2}}

a, b, c, d

G . P

(a + b + c + d)^{2} = (a + b)^{2} + 2 (b + c)^{2} + (b + c)^{2}

\frac{(a + b + c)^{2}}{a^{2} + b^{2} + c^{2}} = \frac{c o t \frac{1}{2} A + c o t \frac{1}{2} B + c o t \frac{1}{2} C}{c o t A + c o t B + c o t C}

If a, b, c are in G.P., prove that:

(i) $a (b^{2} + c^{2}) = c (a^{2} + b^{2})$

(ii) $a^{2} b^{2} c^{2} (\frac{1}{a^{3}} + \frac{1}{b^{3}} + \frac{1}{c^{3}}) = a^{3} + b^{3} + c^{3}$

(iii) $\frac{(a + b + c)^{2}}{a^{2} + b^{2} + c^{2}} = \frac{a + b + c}{a - b + c}$

(iv) $\frac{1}{a^{2} - b^{2}} + \frac{1}{b^{2}} = \frac{1}{b^{2} - c^{2}}$

(v) $(a + 2 b + 2 c) (a - 2 b + 2 c) = a^{2} + 4 c^{2}$ .

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