Question

$3{(\sin \left(x\right)–\cos \left(x\right))}^4+6{(\sin \left(x\right)+\cos \left(x\right))}^2+4\left({\sin }^6\left(x\right)+{\cos }^6\left(x\right)\right)=?$

• A. $11$
• B. $12$
• C. $13$
• D. $14$

Answer 1

The correct option is C

$13$

Explanation for the correct option:

Step-1: Simplify given data:

Given, $3{(\sin \left(x\right)–\cos \left(x\right))}^4+6{(\sin \left(x\right)+\cos \left(x\right))}^2+4\left({\sin }^6\left(x\right)+{\cos }^6\left(x\right)\right)$

$$\begin{gathered} =3{\left[{(\sin \left(x\right)–\cos \left(x\right))}^2\right]}^2+6({\sin }^2\left(x\right)+{\cos }^2\left(x\right)+2\sin \left(x\right)\cos \left(x\right))+4\left({\left({\sin }^2\left(x\right)\right)}^3+{\left({\cos }^2\left(x\right)\right)}^3\right) \\ =3{\left[{\sin }^2\left(x\right)+{\cos }^2\left(x\right)-2\sin \left(x\right)\cos \left(x\right)\right]}^2+6\left(1+2\sin \left(x\right)\cos \left(x\right)\right)+4\left[\left({\sin }^2\left(x\right)\right)+\left({\cos }^2\left(x\right)\right)\left({\sin }^4\left(x\right)+{\cos }^4\left(x\right)-{\sin }^2\left(x\right){\cos }^2\left(x\right)\right)\right] \end{gathered}$$

Step-2: Apply properties ${\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)=1$

$$\begin{gathered} =3{\left[1-2\sin \left(x\right)\cos \left(x\right)\right]}^2+6\left(1+2\sin \left(x\right)\cos \left(x\right)\right)+4\left[1\times \left({\left({\sin }^2\left(x\right)+{\cos }^2\left(x\right)\right)}^2-2{\sin }^2\left(x\right){\cos }^2\left(x\right)-{\sin }^2\left(x\right){\cos }^2\left(x\right)\right)\right] \\ =3\left[1+4{\sin }^2\left(x\right){\cos }^2\left(x\right)-4\sin \left(x\right)\cos \left(x\right)\right]+6+12\sin \left(x\right)\cos \left(x\right)+4\left[1-3{\sin }^2\left(x\right){\cos }^2\left(x\right)\right] \\ =3+12{\sin }^2\left(x\right){\cos }^2\left(x\right)-12\sin \left(x\right)\cos \left(x\right)+6+12\sin \left(x\right)\cos \left(x\right)+4-12{\sin }^2\left(x\right){\cos }^2\left(x\right) \\ =13 \end{gathered}$$

Therefore, correct answer is option $\left(C\right)$