Question

If ${81}^{si{n}^{2}x}+{81}^{co{s}^{2}x}=30$ and $0\le x\le \frac{\pi }{2}$ then $x=$

• A. $\frac{\pi }{6},\frac{\pi }{3}$
• B. $\frac{\pi }{4},\frac{\pi }{2}$
• C. $\frac{3\pi }{4}\frac{2\pi }{3}$
• D. $\frac{\pi }{2}\frac{4\pi }{3}$

Answer 1

The correct option is A $\frac{\pi }{6},\frac{\pi }{3}$

${81}^{si{n}^{2}x}+{81}^{co{s}^{2}x}=30$

$\Rightarrow {81}^{si{n}^{2}x}+{81}^{1-si{n}^{2}x}=30$

$\Rightarrow {81}^{si{n}^{2}x}+\frac{81}{{81}^{si{n}^{2}x}}=30$

$\Rightarrow \frac{({81}^{si{n}^{2}x}{)}^2+81}{{81}^{si{n}^{2}x}}=30$

Let ${81}^{si{n}^{2}x}=y$

$\Rightarrow \frac{y^2+81}{y}=30$

$\Rightarrow y^2-30y+81=0$

$\Rightarrow (y-27)(y-3)=0$

$\Rightarrow y=3,27$

We have ${81}^{si{n}^{2}x}=y$

$\Rightarrow (3^4{)}^{si{n}^{2}x}=y$

$\Rightarrow 3^{4si{n}^{2}x}=y$

When $y=3$

$\Rightarrow 3^{4si{n}^{2}x}=3^1$

$\Rightarrow 4si{n}^{2}x=1$

$\Rightarrow si{n}^{2}x=\frac{1}{4}$

$\Rightarrow sinx=\frac{1}{2}$

$\therefore x=\frac{\pi }{6}$

When $y=27$

$\Rightarrow 3^{4si{n}^{2}x}=3^3$

$\Rightarrow 4si{n}^{2}x=3$

$\Rightarrow si{n}^{2}x=\frac{3}{4}$

$\Rightarrow sinx=\frac{\sqrt{3}}{2}$

$\therefore x=\frac{\pi }{3}$

Hence $x=\frac{\pi }{6},\frac{\pi }{3}$