Question

$(\sin 4\mathrm{\theta })$ can be written as:

• A. $4\sin \theta \left(1–2{\sin }^2\theta \right)\left(\sqrt{1–{\sin }^2\theta }\right)$
• B. $2\sin \theta \cos \theta {\sin }^2\theta$
• C. $4\sin \theta –6{\sin }^3\theta$
• D. None of these

Answer 1

The correct option is A

$4\sin \theta \left(1–2{\sin }^2\theta \right)\left(\sqrt{1–{\sin }^2\theta }\right)$

Explanation for correct option:

Simplifying the equations using trigonometric identities:

Given expression is: $(\sin 4\mathrm{\theta })$

Expanding the expression:

$\sin 4\theta =\sin 2\left(2\theta \right)$

Using the formula $\sin 2A=2\sin A\cos A$ we get:

$\sin 4\theta =2\sin 2\theta \cos 2\theta$

Using the formula $\cos 2\theta =1–2{\sin }^2\theta$ we get:

$$\begin{gathered} \sin 4\theta =2(2\sin \theta \cos \theta )(1–2{\sin }^2\theta ) \\ =4\sin \theta (1–2{\sin }^2\theta )(\cos \theta ) \end{gathered}$$

Using the identity ${\sin }^{2}A+{\cos }^{2}A=1$ we get:

$\sin 4\theta =4\sin \theta \left(1–2{\sin }^2\theta \right)\left(\sqrt{1–s{\text{in}}^2\theta }\right)$

Therefore, the correct answer is option (A).