# JEE Main & Advanced Mathematics Trigonometric Equations Some Important Results

## Some Important Results

Category : JEE Main & Advanced

 (1)      $a=h\,(\cot \alpha -\cot \beta )=\frac{h\sin (\beta -\alpha )}{\sin \alpha .\sin \beta }$     $\therefore \,h=a\sin \alpha \sin \beta \,co\text{sec }(\beta -\alpha )$   and   $d=h\cot \beta =a\sin \alpha .\cos \beta .\text{cosec}(\beta -\alpha )$ (2)      $H=x\cot \alpha \tan (\alpha +\beta )$ (3)        $a=h(\cot \alpha +\cot \beta ),$ where by     $h=a\sin \alpha .\sin \beta .\text{cosec}(\alpha +\beta )$ and     $d=h\cot \beta =a\sin \alpha .\cos \beta .\text{cosec }(\alpha +\beta )$ (4)    $H=\frac{h\cot \beta }{\cot \alpha }$ (5)      $h=\frac{H\sin (\beta -\alpha )}{\cos \alpha \sin \beta }$   or $H=\frac{h\cot \alpha }{\cot \alpha -\cot \beta }$ (6)      $H=\frac{a\sin (\alpha +\beta )}{\sin (\beta -\alpha )}$ (7)      $AB=CD$. Then, $x=y\tan \left( \frac{\alpha +\beta }{2} \right)$ (8)      $h=\frac{d}{\sqrt{{{\cot }^{2}}\beta +{{\cot }^{2}}\alpha }}$ (9)      $h=\frac{AB}{\sqrt{{{\cot }^{2}}\beta -{{\cot }^{2}}\alpha }}$ (10)    $h=AP\sin \alpha$   $=a\sin \alpha .\sin \gamma .c\text{osec}(\beta -\gamma )$ and If $AQ=d$, then $d=AP\text{ }\cos \alpha =a\text{ }\cos \alpha .\sin \gamma \text{ }.\text{ cosec }(\beta -\gamma )$ (11)        $AP=a\sin \gamma .co\text{sec}(\alpha -\gamma )$,   $AQ=a\sin \delta .\text{cosec}(\beta -\delta )$   and apply,   $P{{Q}^{2}}=A{{P}^{2}}+A{{Q}^{2}}-2AP.AQ\cos \theta$

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