Question

Find x from the following equations:

$\left(i\right)cosec({90}^{\circ }+\theta )+xcos\theta cot({90}^{\circ }+\theta )$

$=sin({90}^{\circ }+\theta )$

$\left(ii\right)xcot({90}^{\circ }+\theta )+tan({90}^{\circ }+\theta )sin\theta +cosec({90}^{\circ }+\theta )=0$

Answer 1

$\left(i\right)cosec({90}^{\circ }+\theta )+xcos\theta cot({90}^{\circ }+\theta )$

$=sin({90}^{\circ }+\theta )$

$\Rightarrow sec\theta +xcos\theta x(-tan\theta )=cos\theta$

$\Rightarrow \frac{1}{cos\theta }+xcos\theta x\frac{(-sin\theta )}{cos\theta }=cos\theta$

$\Rightarrow \frac{1}{cos\theta }-xsin\theta =cos\theta$

$\Rightarrow \frac{1-xsin\theta cos\theta }{cos\theta }=cos\theta$

$\Rightarrow 1-xsin\theta cos\theta =co{s}^2\theta$

$\Rightarrow 1-co{s}^2\theta =xsin\theta cos\theta$

$\Rightarrow si{n}^2\theta =xsin\theta cos\theta$

$\Rightarrow sin\theta =xcos\theta$

$\Rightarrow x=\frac{sin\theta }{cos\theta }$

$=tan\theta$

Hence $x=tan\theta$

(ii)We have $xcot({90}^{\circ }+\theta )+tan({90}^{\circ }+\theta )sin\theta +cosec({90}^{\circ }+\theta )=0$

$\Rightarrow x(-tan\theta )-cot\theta \times sin\theta +sec\theta =0$

$\Rightarrow -xtan\theta -\frac{cos\theta }{sin\theta }\times sin\theta +\frac{1}{cos\theta }=0$

$\Rightarrow -x\frac{sin\theta }{cos\theta }-cos\theta +\frac{1}{cos\theta }=0$

$\Rightarrow \frac{-xsin\theta -co{s}^2\theta +1}{cos\theta }=0$

$\Rightarrow -xsin\theta +1-co{s}^2\theta =0$

$\Rightarrow -xsin\theta +si{n}^2\theta =0$

$\Rightarrow xsin\theta =si{n}^2\theta$

$\Rightarrow x=\frac{si{n}^2\theta }{sin\theta }$

$\Rightarrow x=sin\theta$