If 8 = 15, find(i) , (ii) , (iii) ^2 - ^2 - Vidyadip

Question

If $8 \tan\theta = 15$, find$(i) \sin\theta , (ii) \cot\theta , (iii) \sin^2\theta - \cot^2\theta$

✓

Answer

$8 \tan \theta=15$
$\Rightarrow \tan \theta=\frac{15}{8}=\frac{\text { Perpendicular }}{\text { Base }}$
Hypotenuse
$=\sqrt{(\text { Perpendicular })^2+(\text { Base })^2} $
$=\sqrt{15^2+8^2} $
$=\sqrt{225+64} $
$=\sqrt{289} $
$=17$
$ii) \sin \theta=\frac{\text { Perpendicular }}{\text { Hypotenuse }}=\frac{15}{17}$
$\cot \theta=\frac{1}{\tan \theta}=\frac{8}{15}$
$iii) \sin ^2 \theta-\cot ^2 \theta$
$=(\sin \theta+\cot \theta)(\sin \theta-\cot \theta) $
$=\left(\frac{15}{17}+\frac{8}{15}\right)\left(\frac{15}{17}-\frac{8}{15}\right) $
$=\left(\frac{225+136}{225}\right)\left(\frac{225-136}{225}\right) $
$=\left(\frac{361}{225}\right)\left(\frac{89}{255}\right) $
$=\frac{32129}{65025} .$

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