\[\LARGE{\color {purple} {TRIGONOMETRIC\ IDENTITIES}}\]
\[\color {royalblue} {Fundamental\ Trigonometrical\ identities}:\ \hspace{20cm}\]
\[sin^2θ\ +\ cos^2θ\ =\ 1\ \hspace{10cm}\]
\[1\ +\ tan^2θ\ =\ sec^2θ\ \hspace{10cm}\]
\[1\ +\ cot^2θ\ =\ cosec^2θ\ \hspace{10cm}\]
\[\color {royalblue} {Trigonometrical\ ratios\ of\ known\ angles}:\ \hspace{20cm}\]
\[Sin ( A + B )\ =\ Sin A\ Cos B\ +\ Cos A\ Sin B\]
\[Sin ( A – B )\ =\ Sin A\ Cos B\ -\ Cos A\ Sin B\]
\[Cos( A + B )\ =\ Cos A\ Cos B\ -\ Sin A Sin B\]
\[Cos( A – B )\ =\ Cos A\ Cos B\ +\ Sin A Sin B\]
\[Tan(A + B)\ =\ \frac{Tan A\ +\ Tan B}{1\ -\ Tan A\ Tan B}\]
\[Tan(A – B)\ =\ \frac{Tan A\ -\ Tan B}{1\ +\ Tan A\ Tan B}\]
\[\color {brown} {Note}:\ Sin\ ( – θ )\ =\ -\ sin\ θ\ \hspace{2cm}\ cos\ ( – θ )\ =\ cos\ θ\]
\[\color {royalblue} {Multiple\ Angles\ of\ 2A}:\ \hspace{20cm}\]
\[1\ (i)\ Sin\ 2A\ =\ 2\ Sin\ A\ Cos\ A\ \hspace{5cm}\ (ii)\ Sin\ 2A\ =\ \frac{2\ Tan\ A}{1\ +\ Tan^2\ A}\]
\[2\ (i)\ Cos\ 2A\ =\ Cos^2A\ -\ Sin^2A\ \hspace{5cm}\ (ii)\ Cos\ 2A\ =\ \frac{1\ -\ Tan^2A}{1\ +\ Tan^2\ A}\]
\[3.\ Tan\ 2A\ =\ \frac{2\ Tan\ A}{1\ -\ Tan^2\ A}\ \hspace{10cm}\]
\[4.\ Sin^2A\ =\ \frac{1\ -\ Cos\ 2A}{2}\ \hspace{5cm}\ Note:\ 1\ -\ 2\ Sin^2A\ =\ Cos\ 2A\]
\[5.\ Cos^2A\ =\ \frac{1\ +\ Cos\ 2A}{2}\ \hspace{10cm}\]
\[6.\ Tan^2A\ =\ \frac{1\ -\ Cos\ 2A}{1\ +\ Cos\ 2A}\ \hspace{10cm}\]
\[\color {royalblue} {Multiple\ Angles\ of\ 3A}:\ \hspace{20cm}\]
\[1.\ Sin\ 3\ \theta\ =\ 3\ Sin\ \theta\ -\ 4\ Sin^3\ \theta\ \hspace{10cm}\]
\[2.\ Cos\ 3\ \theta\ =\ 4\ Cos^3\ \theta\ -\ 3\ Cos\ \theta\ \hspace{10cm}\]
\[3.\ tan\ 3\ \theta\ =\ \frac{3\ tan\ \theta\ -\ tan^3\ \theta}{1\ -\ 3\ tan^2\ \theta}\ \hspace{10cm}\]
\[\LARGE{\color {purple} {INVERSE\ TRIGONOMETRIC\ FUNCTIONS}}\]
\[\color {royalblue} {Sum\ and\ Product\ Identities}:\ \hspace{20cm}\]
\[1.\ Sin\ A\ +\ Sin\ B =\ 2\ Sin\ (\frac{A\ +\ B}{2})\ Cos\ (\frac{A\ -\ B}{2})\ \hspace{10cm}\]
\[2.\ Sin\ A\ -\ Sin\ B =\ 2\ Cos\ (\frac{A\ +\ B}{2})\ Sin\ (\frac{A\ -\ B}{2})\ \hspace{10cm}\]
\[3.\ Cos\ A\ +\ Cos\ B =\ 2\ Cos\ (\frac{A\ +\ B}{2})\ Cos\ (\frac{A\ -\ B}{2})\ \hspace{10cm}\]
\[4.\ Cos\ A\ -\ Cos\ B =\ -\ 2\ Sin\ (\frac{A\ +\ B}{2})\ Sin\ (\frac{A\ -\ B}{2})\ \hspace{10cm}\]
\[5.\ Sin\ A\ Cos\ B\ =\ \frac{1}{2}[Sin\ (A\ +\ B)\ +\ Sin\ (A\ -\ B)]\ \hspace{10cm}\]
\[7.\ Cos\ A\ Cos\ B\ =\ \frac{1}{2}[Cos\ (A\ +\ B)\ +\ Cos\ (A\ -\ B)]\ \hspace{10cm}\]
\[8.\ Sin\ A\ Sin\ B\ =\ \frac{-\ 1}{2}[Cos\ (A\ +\ B)\ -\ Cos\ (A\ -\ B)]\ \hspace{10cm}\]
\[9.\ Sin\ (A\ +\ B)Sin\ (A\ -\ B)\ =\ Sin^2\ A\ -\ Sin^2\ B\ \hspace{10cm}\]
\[10.\ Cos\ (A\ +\ B)Cos\ (A\ -\ B)\ =\ Cos^2\ A\ -\ Sin^2\ B\ \hspace{10cm}\]
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