Chapter 5.1, Problem 50E

To determine

To simplify: The given trigonometric expression and check the result using a graphing utility.

Answer to Problem 50E

  $1-2{\sin }^{2}x+{\sin }^{4}x={\cos }^{4}x$

The result is graphically verified as the curve of the given expression coincides at all points with the graph of its result.

Explanation of Solution

Given information:

The given trigonometric expression:

  $1-2{\sin }^{2}x+{\sin }^{4}x$

Formula used:

The square of a difference can be written as:

  ${(a-b)}^2=a^2-2ab+b^2$

Calculation:

As per problem,

The given trigonometric expression

  $1-2{\sin }^{2}x+{\sin }^{4}x$

The expression has the polynomial form $a^2-2ab+b^2$ and can be factorized as:

  $\begin{array}{}$ 1-2{\sin }^{2}x+{\sin }^{4}x=1-2{\sin }^{2}x+{({\sin }^{2}x)}^2\text{ [Rule of exponents]}$\end{array}$

  $\text{ = }{(1)}^2-2\cdot 1\cdot {\sin }^{2}x+{(-{\sin }^{2}x)}^2\text{ [Write in }{a}^2-2ab+b^2\text{ form]}$                              =  (1−sin2x)2                          [Use (a−b)2 formula]                              =  (cos2x)2                          [Pythagorean identity: cos2x=1−sin2x ]                              = cos4x                               [ Rule of exponents]

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions ${\text{Y}}_1=1-2{\sin }^{2}x+{\sin }^{4}x$ and ${\text{Y}}_2={\cos }^{4}x$ , which is required to graph.

Step 4: Press GRAPH button to graph the function.

  

The curve of ${\text{Y}}_1$ is indicated by yellow line and the curve of ${\text{Y}}_2$ is indicated by black line. Both the curves coincide.

Interpretation: According to the above graph, it can be observed that ${\text{Y}}_1={\text{Y}}_2$ as both the graphs coincide.

Therefore, from the graph, $1-2{\sin }^{2}x+{\sin }^{4}x={\cos }^{4}x$ , which is same as calculated. Thus, the result is graphically verified.