Question

$\frac{si{n}^6\theta +co{s}^6\theta -1}{si{n}^4\theta +co{s}^4\theta -1}$ equals

• A. $\frac{2}{3}$
• B. 1
• C. $\frac{3}{2}$
• D. 2

Answer 1

The correct option is C $\frac{3}{2}$

$si{n}^4\theta +co{s}^4\theta =(si{n}^2\theta +co{s}^2\theta {)}^2-2si{n}^2\theta co{s}^2\theta$

$\Rightarrow si{n}^4\theta +co{s}^4\theta =1-2si{n}^2\theta co{s}^2\theta$ ......(i) [Q $si{n}^2\theta +co{s}^2\theta =1$]

Also, $si{n}^6\theta +co{s}^6\theta =(si{n}^2\theta {)}^3+(co{s}^2\theta {)}^3$

$=(si{n}^2\theta +co{s}^2\theta )\left[\right(si{n}^2\theta {)}^2+(co{s}^2\theta {)}^2-si{n}^2\theta co{s}^2\theta ]$

$=1\times \left[\right(si{n}^2\theta +co{s}^2\theta {)}^2-3si{n}^2\theta co{s}^2\theta ]$

$=1-3si{n}^{\theta }co{s}^2\theta$

i.e., $si{n}^6\theta +co{s}^6\theta =1-3si{n}^2\theta co{s}^2\theta$.....(ii)

$\therefore \frac{si{n}^6\theta +co{s}^6\theta -1}{si{n}^4\theta +co{s}^4\theta -1}=\frac{-3si{n}^2\theta co{s}^2\theta }{-2si{n}^2\theta co{s}^2\theta }=\frac{3}{2}$