Question

Find the value $si{n}^4\theta$ - $co{s}^4\theta$ + 2$co{s}^2\theta$, when $\theta =\frac{\Pi }{7}$

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Answer 1

Before we actually solve this, let's find out the value of this expression for some known angles for us.

$\theta$ = 0

F(0) = $si{n}^4\theta$ - $co{s}^4\theta$ + 2$co{s}^2\theta$ = 0 - 1 + 2 = 1

$\theta =\frac{\Pi }{2}$

F ($\frac{\Pi }{2}$) = 1 - 0 + 2 $\times$ 0 = 1

$\theta$ = $\frac{\Pi }{4}$

F ($\frac{\Pi }{2}$) = $\frac{1}{4}$ - $\frac{1}{4}$ + 2 $\times$ $\frac{1}{2}$ = 1

We see that the value of the expression remains the same for different $\theta$, we considered. The motivation for doing this is $\theta$ = $\theta$ = $\frac{\Pi }{7}$ is not a standard / commonly used angle. So the assumption is that the expression will be independent of $\theta$ (We can't be sure about this)

We have to simplify / modify the expression. One way of proceeding is by replacing $si{n}^2\theta$ or $co{s}^2\theta$ with 1 - $co{s}^2\theta$ or 1 - $si{n}^2\theta$; another way of proceeding is by taking $co{s}^2\theta$ from the last two terms. If we proceed this was, it won't simplify the expression a lot. So we will go with replacing $si{n}^2\theta$ with 1 - $co{s}^2\theta$.

$\Rightarrow$ $si{n}^4\theta$ - $co{s}^4\theta$ + 2$co{s}^2\theta$ = $(1-co{s}^2\theta {)}^2$ - $co{s}^4\theta$ + 2 $co{s}^2\theta$

= 1 + $co{s}^4\theta$ - 2 $co{s}^2\theta$ - $co{s}^4\theta$ + 2 $co{s}^2\theta$

= 1

As we expected, the expression is independent of $\theta$.

Key steps:

(1) Guessing that the expression could be independent of $\theta$

(2) Attempting to modify the expression using basic identities ($co{s}^2\theta$ + $si{n}^2\theta$ = 1)