Multiples And Sub-Multiples
We know
sin2A = 2 sinA cosA
put A =\(
\frac{A}
{2}
\)
sin 2\(
\left( {\frac{A}
{2}} \right)
\)=2sin\(
\frac{A}
{2}
\).cos\(
\frac{A}
{2}
\)
sinA = 2sin\(
\frac{A}
{2}
\). cos\(
\frac{A}
{2}
\) ....(a)
We know
\(
\sin 2A = \frac{{2\tan A}}
{{1 + \tan ^2 A}}
\)
Put A=\(
\frac{A}
{2}
\)
\(
\sin \left( {2.\frac{A}
{2}} \right) = \frac{{2\tan \frac{A}
{2}}}
{{1 + \tan ^2 \frac{A}
{2}}}
\)
\(
\sin A = \frac{{2\tan \frac{A}
{2}}}
{{1 + \tan ^2 \frac{A}
{2}}}
\) ……...(b)
from a and b
\(
\sin A = 2\sin \frac{A}
{2}\,\,\cos \frac{A}
{2} = \frac{{2\tan \frac{A}
{2}}}
{{1 + \tan ^2 \frac{A}
{2}}}\,\,\,\,\,\, \to (ix)
\)
We know
\(
\cos 2A = \cos ^2 A - \sin ^2 A = 1 - 2\sin ^2 A = 2\cos ^2 A - 1 = \frac{{1 - \tan ^2 A}}
{{1 + \tan ^2 A}}
\)
Put A=\(
\frac{A}
{2}
\)
\(
\cos A = \cos ^2 \frac{A}
{2} - \sin ^2 \frac{A}
{2} = 1 - 2\sin ^2 \frac{A}
{2} = 2\cos ^2 \frac{A}
{2} - 1 = \frac{{1 - \tan ^2 \frac{A}
{2}}}
{{1 + \tan ^2 \frac{A}
{2}}}\,\,\,\,\,\,\,\,\,\, \to (x)
\)
we know
tan 2A=\(
\frac{{2\tan A}}
{{1 - \tan ^2 A}}
\)
put A= \(
\frac{A}
{2}
\)
\(
\tan A = \frac{{2\tan \frac{A}
{2}}}
{{1 - \tan ^2 \frac{A}
{2}}}\,\,\,\, \to (xi)
\)
We know
\(
\cot 2A = \frac{{\cot ^2 A - 1}}
{{2\cot A}}
\)
Put A=\(
\frac{A}
{2}
\)
\(
\cot A = \frac{{\cot ^2 \frac{A}
{2} - 1}}
{{2\cot \frac{A}
{2}}}\,\,\,\, \to (xii)
\)