Question

The value of the expression ${\sin }^6\left(\theta \right)+{\cos }^6\left(\theta \right)+3{\sin }^2\left(\theta \right)·{\cos }^2\left(\theta \right)$ is

• A. $0$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

The correct option is C

$1$

Explanation for correct option :

Given trigonometric expression is ${\sin }^6\left(\theta \right)+{\cos }^6\left(\theta \right)+3{\sin }^2\left(\theta \right)·{\cos }^2\left(\theta \right)$

$$\begin{gathered} \therefore {\sin }^6\left(\theta \right)+{\cos }^6\left(\theta \right)+3{\sin }^2\left(\theta \right)·{\cos }^2\left(\theta \right) \\ ={\left({\sin }^2\left(\theta \right)\right)}^3+{\left({\cos }^2\left(\theta \right)\right)}^3+3\left[\left({\sin }^2\left(\theta \right)·{\cos }^2\left(\theta \right)\right)\left({\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)\right)\right]\left[\because {\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)=1\right] \\ ={\left({\sin }^2\left(\theta \right)\right)}^3+{\left({\cos }^2\left(\theta \right)\right)}^3+3\left({\sin }^4\left(\theta \right)·{\cos }^2\left(\theta \right)\right)+3\left({\sin }^2\left(\theta \right)·{\cos }^4\left(\theta \right)\right) \\ ={\left({\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)\right)}^3 \\ ={\left(1\right)}^3 \\ =1 \end{gathered}$$

Hence, option (C) is correct.