Question

If sin x + cosec x = 3, then find the value of $\dfrac{1+{{\sin }^{4}}x}{{{\sin }^{2}}x}$.

Answer 1

Hint: We will use the concept of $\cos ecA=\dfrac{1}{\sin A}$ to get the value of $\dfrac{1+{{\sin }^{4}}x}{{{\sin }^{2}}x}$ , $\operatorname{cosec}x$ is reciprocal of sin$x$. Then, we will try to get the expression of $\dfrac{1+{{\sin }^{4}}x}{{{\sin }^{2}}x}$ by solving and re -arranging the equation after the substation of $\cos ecA=\dfrac{1}{\sin A}$ . Also, we need to use formula of ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ .

Complete step-by-step answer:
Now, we know that $\cos ecA$ is reciprocal of \[\sin A\] that is \[\cos ecA=\dfrac{1}{\sin A}\] .
So, we can write $\sin x+\cos ecx=\sin x+\dfrac{1}{\sin x}$.
Then, $\sin x+\dfrac{1}{\sin x}=3$ .
Taking L.C.M of $\sin x$ and $\dfrac{1}{\sin x}$,
$\dfrac{{{\sin }^{2}}x+1}{\sin x}=3$ .
Taking $\sin x$ in denominator on left hand side to the numerator of right hand side using cross multiplication,
${{\sin }^{2}}x+1=3\times \sin x$ ……( i )
Now, squaring both sides in equation ( i ), we get
${{\sin }^{4}}x+2{{\sin }^{2}}x+1=9{{\sin }^{2}}x$ , as ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$
Taking $2{{\sin }^{2}}x$ from left hand side to right hand side, we get
${{\sin }^{4}}x+1=9{{\sin }^{2}}x-2{{\sin }^{2}}x$,
On simplifying by subtracting $2{{\sin }^{2}}x$ from $9{{\sin }^{2}}x$ , we get
${{\sin }^{4}}x+1=7{{\sin }^{2}}x$,
Taking ${{\sin }^{2}}x$ in numerator on right hand side to the denominator of left hand side using cross multiplication ,
$\dfrac{1+{{\sin }^{4}}x}{{{\sin }^{2}}x}=7$, which is equals to same expression mentioned in the question.
Hence, the answer we obtain is equals to 7 .

Note:You can also solve the equation ${{\sin }^{2}}x+1=3\times \sin x$by reducing it into quadratic equation by letting $\sin x$ = t which will give us quadratic equation ${{t}^{2}}-3t+1=0$ which can further be solved for value of sin x and then we can substitute the obtained value in expression $\dfrac{1+{{\sin }^{4}}x}{{{\sin }^{2}}x}$ to get it is value. Always remember that $\cos ecA=\dfrac{1}{\sin A}$. Cross multiplication should be done in such a way that you get some expression which is given in question or it can be reduced to the same expression in question.