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If \[0\le x\le \pi \] and ${{81}^{{{\sin }^{2}}x}}+{{81}^{{{\cos }^{2}}x}}=30$, then $x$ is equal
(a) $\dfrac{\pi }{6}$
(b) $\dfrac{\pi }{3}$
(c) $\dfrac{5\pi }{6}$
(d) $\dfrac{2\pi }{3}$
(e) All correct

Last updated date: 16th Jul 2024
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Since the question contains both ${{\sin }^{2}}x$ and ${{\cos }^{2}}x$ , we will write ${{\cos}^{2}}x=1-{{\sin }^{2}}x$ so that we can get ${{81}^{{{\sin }^{2}}x}}$ in both the terms in left hand side of the equation given in the question. Then substitute ${{81}^{{{\sin }^{2}}x}}=t$ and solve the obtained quadratic equation.

In the question, we are given an equation ${{81}^{{{\sin }^{2}}x}}+{{81}^{{{\cos}^{2}}x}}=30.........(1)$.
We have a trigonometric identity,
& {{\sin }^{2}}x+{{\cos }^{2}}x=1 \\
& \Rightarrow {{\cos }^{2}}x=1-{{\sin }^{2}}x..........\left( 2 \right) \\
Substituting ${{\cos }^{2}}x=1-{{\sin }^{2}}x$ from equation $\left( 2 \right)$ in equation $\left( 1
\right)$, we get,
${{81}^{{{\sin }^{2}}x}}+{{81}^{1-{{\sin }^{2}}x}}=30...........\left( 3 \right)$
We have a property of exponents,
${{a}^{b-c}}=\dfrac{{{a}^{b}}}{{{a}^{c}}}$, where $a,b,c\in R$.
Using this property in equation $\left( 3 \right)$ with $a=81,b=1$ and $c={{\sin }^{2}}x$, we get,
${{81}^{{{\sin }^{2}}x}}+\dfrac{{{81}^{1}}}{{{81}^{{{\sin }^{2}}x}}}=30...........\left( 4 \right)$
Now, let us assume ${{81}^{{{\sin }^{2}}x}}=t$. Substituting ${{81}^{{{\sin }^{2}}x}}=t$ in equation
$\left( 4 \right)$, we get,

& \Rightarrow \dfrac{{{t}^{2}}+81}{t}=30 \\
& \Rightarrow {{t}^{2}}+81=30t \\
& \Rightarrow {{t}^{2}}-30t+81=0 \\
Now, we have got a quadratic equation which can be easily solved by using the method of splitting the middle term. Solving this quadratic equation,
& {{t}^{2}}-3t-27t+81=0 \\
& \Rightarrow t\left( t-3 \right)-27\left( t-3 \right)=0 \\
& \Rightarrow \left( t-3 \right)\left( t-27 \right)=0 \\
$\Rightarrow t=27$ or $t=3.........\left( 5 \right)$
Since we have earlier assumed ${{81}^{{{\sin }^{2}}x}}=t$, we can now again substitute
$t={{81}^{{{\sin }^{2}}x}}$ in equation $\left( 5 \right)$. Substituting $t={{81}^{{{\sin }^{2}}x}}$ in
equation $\left( 5 \right)$, we get,
${{81}^{{{\sin }^{2}}x}}=27$ or ${{81}^{{{\sin }^{2}}x}}=3$
Since, the above two equation contains exponential terms, in order to solve them, we first have to
make their base equal. We know $81={{3}^{4}}$ and $27={{3}^{3}}$. Hence, substituting
$81={{3}^{4}}$ and $27={{3}^{3}}$ in the above equation, we get,
${{\left( {{3}^{4}} \right)}^{{{\sin }^{2}}x}}={{3}^{3}}$ or ${{\left( {{3}^{4}} \right)}^{{{\sin }^{2}}x}}=3$
$\Rightarrow {{3}^{4{{\sin }^{2}}x}}={{3}^{3}}$ or ${{3}^{4}}^{{{\sin }^{2}}x}=3............\left( 6 \right)$
Now, the base of all the exponential terms in the above two equations in equation $\left( 6 \right)$ is the same. So, we can directly equate the powers of the terms on both the sides of equality.
$\Rightarrow 4{{\sin }^{2}}x=3$ or $4{{\sin }^{2}}x=1$
$\Rightarrow {{\sin }^{2}}x=\dfrac{3}{4}$ or ${{\sin }^{2}}x=\dfrac{1}{4}$
$\Rightarrow \sin x=\pm \dfrac{\sqrt{3}}{2}$ or $\sin x=\pm \dfrac{1}{2}.........\left( 7 \right)$
In the question, it is given that \[0\le x\le \pi \]. For this domain, the value of $\sin x$ is positive .
So, we can neglect $\sin x=-\dfrac{\sqrt{3}}{2}$ and $\sin x=-\dfrac{1}{2}$
as the solution of the equation $\left( 7 \right)$.
Removing $\sin x=-\dfrac{\sqrt{3}}{2}$ and $\sin x=-\dfrac{1}{2}$ from equation $\left( 7 \right)$, we
$\sin x=+\dfrac{\sqrt{3}}{2}$ or $\sin x=+\dfrac{1}{2}$
Solving the above equation in the domain \[0\le x\le \pi \], we get,
$x=\dfrac{\pi }{3},\dfrac{2\pi }{3}$ or $x=\dfrac{\pi }{6},\dfrac{5\pi }{6}$.

So, the possible values of $x$ are $\dfrac{\pi }{3},\dfrac{\pi }{6},\dfrac{2\pi }{3},\dfrac{5\pi }{6}$.
Therefore the final answer is option (e).

Note: There is a possibility that one may commit a mistake if he/she does not consider $\dfrac{2\pi}{3}$ and $\dfrac{5\pi }{6}$ as possible solutions of the equation. But as it is mentioned in the question that \[0\le x\le \pi \], we have to consider $\dfrac{2\pi }{3}$ and $\dfrac{5\pi }{6}$ as possible solutions of the equation.