Chapter 5.3, Problem 10E

a.

To determine

To verify: that the x -value is a solution of the equation ${\csc }^{4}x-4{\csc }^{2}x=0$

Explanation of Solution

Given:

  $x=\frac{\pi }{6}$

Proof:

The given equation is ${\csc }^{4}x-4{\csc }^{2}x=0$

Substituting $x=\frac{\pi }{6}$ in this equation

  $\begin{array}{l}{\csc }^4\left(\frac{\pi }{6}\right)-4{\csc }^2\frac{\pi }{6}=0\\ \Rightarrow {\left(2\right)}^4-4\cdot {\left(2\right)}^2=0\\ \Rightarrow 16-16=0\\ \Rightarrow 0=0\Rightarrow \text{True}\end{array}$

Since, the value of x satisfies the given equation.

Hence, the given x -value is a solution of the equation.

b.

To determine

To verify: that the x -value is a solution of the equation ${\csc }^{4}x-4{\csc }^{2}x=0$

Explanation of Solution

Given:

  $x=\frac{5\pi }{6}$

Proof:

The given equation is ${\csc }^{4}x-4{\csc }^{2}x=0$

Substituting $x=\frac{5\pi }{6}$ in this equation

  $\begin{array}{l}{\csc }^4\left(\frac{5\pi }{6}\right)-4{\csc }^2\frac{5\pi }{6}=0\\ \Rightarrow {\left(2\right)}^4-4\cdot {\left(2\right)}^2=0\\ \Rightarrow 16-16=0\\ \Rightarrow 0=0\Rightarrow \text{True}\end{array}$

Since, the value of x satisfies the given equation.

Hence, the given x -value is a solution of the equation.