Question

Use the suitable identity and simplify the given expression.$2(si{n}^6\theta +co{s}^6\theta )-3(si{n}^4\theta +co{s}^4\theta )+1$

• A. 0
• B. 1
• C. 2
• D. 6

Answer 1

The correct option is A

0

Given expression:
$2(si{n}^6\theta +co{s}^6\theta )-3(si{n}^4\theta +co{s}^4\theta )+1$

$=2\left[\right(si{n}^2\theta {)}^3+\left(co{s}^2\theta {)}^3\right]-3\left[\right(si{n}^2\theta +co{s}^2\theta {)}^2-2si{n}^2\theta co{s}^2\theta ]+1$

We know that, $a^3+b^3=(a+b{)}^3-3ab(a+b)$
By applying this formula, we get,$=2\left[\right(si{n}^2\theta +co{s}^2\theta {)}^3-3si{n}^2\theta co{s}^2\theta (si{n}^2\theta +co{s}^2\theta )]-3[(si{n}^2\theta +co{s}^2\theta {)}^2-2si{n}^2\theta co{s}^2\theta ]+1$

Using the identity, $si{n}^2\theta +co{s}^2\theta =1,$we get,
$=2[1-3si{n}^2\theta co{s}^2\theta ]-3[1-2si{n}^2\theta co{s}^2\theta ]+1$

$=2-6si{n}^2\theta co{s}^2\theta -3+6si{n}^2\theta co{s}^2\theta +1$

$=3-3$

$=0$