Chapter 5.2, Problem 42E

To determine

To complete the given table using graphing utility:

x	0.2	0.4	0.6	0.8	1.0	1.2	1.4
$y_1$
$y_2$

And using graph and completed table prove $y_1=y_2$ , where

  $y_1=\left(1+{\tan }^{2}x\right){\sin }^{2}x,y_2={\tan }^{2}x$

Answer to Problem 42E

x	0.2	0.4	0.6	0.8	1.0	1.2	1.4
$y_1$	0.041	0.179	0.468	1.06	2.426	6.616	33.615
$y_2$	0.041	0.179	0.468	1.06	2.426	6.616	33.615

Explanation of Solution

Given:

  $y_1=\left(1+{\tan }^{2}x\right){\sin }^{2}x,y_2={\tan }^{2}x$

Calculation:

To complete the given table first draw the graphs of functions $y_1,\text{ and }{y}_2$ using graphing tool and check the values of graph at given points and note them down in their corresponding places.

Graph of function $y_1=\left(1+{\tan }^{2}x\right){\sin }^{2}x$ is

  

And graph of function $y_2={\tan }^{2}x$ is

  

Now, complete the given table using the above graphs by filling the value of respective functions at given points:

x	0.2	0.4	0.6	0.8	1.0	1.2	1.4
$y_1$	0.041	0.179	0.468	1.06	2.426	6.616	33.615
$y_2$	0.041	0.179	0.468	1.06	2.426	6.616	33.615

Now, note from above that graph of both functions $y_1,\text{ and }{y}_2$ are similar and also the value of both functions at given points is same.

Thus, it is proved that $y_1=y_2$ .