Question

# If $\mathrm{cot}\mathrm{\theta }-\mathrm{tan}\mathrm{\theta }=\mathrm{sec}\mathrm{\theta }$, then, θ is equal to (a) $2n\pi +\frac{3\pi }{2},n\in Z$ (b) $n\pi +{\left(-1\right)}^{n}\frac{\pi }{6},n\pi Z$ (c) $n\pi +\frac{\pi }{2},n\in Z$ (d) none of these.

Open in App
Solution

## (b) $n\pi +{\left(-1\right)}^{n}\frac{\pi }{6},n\in Z$ Given equation: $cot\theta -\mathrm{tan}\theta =sec\theta \phantom{\rule{0ex}{0ex}}⇒\frac{\mathrm{cos}\theta }{\mathrm{sin}\theta }-\frac{\mathrm{sin}\theta }{\mathrm{cos}\theta }=\frac{1}{\mathrm{cos}\theta }\phantom{\rule{0ex}{0ex}}⇒\frac{{\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta }{\mathrm{sin}\theta \mathrm{cos}\theta }=\frac{1}{\mathrm{cos}\theta }\phantom{\rule{0ex}{0ex}}⇒{\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta =\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}⇒\left(1-{\mathrm{sin}}^{2}\theta \right)-{\mathrm{sin}}^{2}\theta =\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}⇒1-2{\mathrm{sin}}^{2}\theta =\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}⇒2{\mathrm{sin}}^{2}\theta +\mathrm{sin}\theta -1=0\phantom{\rule{0ex}{0ex}}⇒2{\mathrm{sin}}^{2}\theta +2\mathrm{sin}\theta -\mathrm{sin}\theta -1=0\phantom{\rule{0ex}{0ex}}⇒2\mathrm{sin}\theta \left(\mathrm{sin}\theta +1\right)-1\left(\mathrm{sin}\theta +1\right)=0\phantom{\rule{0ex}{0ex}}⇒\left(\mathrm{sin}\theta +1\right)\left(2\mathrm{sin}\theta -1\right)=0$ $⇒\mathrm{sin}\theta +1=0$ or $2\mathrm{sin}\theta -1=0$ $⇒\mathrm{sin}\theta =-1$ or $\mathrm{sin}\theta =\frac{1}{2}$ Now, $\mathrm{sin}\theta =-1⇒\mathrm{sin}\theta =\mathrm{sin}\frac{3\mathrm{\pi }}{2}⇒\theta =m\mathrm{\pi }+\left(-1{\right)}^{m}\frac{3\mathrm{\pi }}{2},m\in Z\phantom{\rule{0ex}{0ex}}$ And, $\mathrm{sin}\theta =\frac{1}{2}⇒\mathrm{sin}\theta =\mathrm{sin}\frac{\mathrm{\pi }}{6}⇒\theta =n\mathrm{\pi }+\left(-1{\right)}^{n}\frac{\mathrm{\pi }}{6},n\in Z$ ∴ $\theta =n\mathrm{\pi }+\left(-1{\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{6},\mathrm{n}\in \mathrm{Z}$

Suggest Corrections
0
Join BYJU'S Learning Program
Select...
Related Videos
Property 5
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
Select...