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Select ALL of the correct answers. *

2sin^(4)theta+2sin^(2)theta*cos^(2)theta=1

$5$ . Select ALL of the correct answers. * $\newline$ $2 \sin ^{4} \theta+2 \sin ^{2} \theta \cdot \cos ^{2} \theta=1$

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Product property of logarithms

Full solution

Q.

5

. Select ALL of the correct answers. *

\newline

2 \sin ^{4} \theta+2 \sin ^{2} \theta \cdot \cos ^{2} \theta=1

Apply Pythagorean identity: Use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ to simplify the equation. $\newline$ $2\sin^4(\theta) + 2\sin^2(\theta)(1 - \sin^2(\theta)) = 1$
Distribute terms: Distribute the $2\sin^{2}\theta$ into the parentheses. $\newline$ $2\sin^{4}\theta + 2\sin^{2}\theta - 2\sin^{4}\theta = 1$
Combine like terms: Combine like terms. $2\sin^{4}\theta - 2\sin^{4}\theta + 2\sin^{2}\theta = 1$
Cancel out terms: The terms $2\sin^{4}\theta$ cancel each other out. $\newline$ $2\sin^{2}\theta = 1$
Isolate $\sin^2(\theta)$ : Divide both sides by $2$ to isolate $\sin^{2}\theta$ . $\newline$ $\sin^{2}\theta = \frac{1}{2}$
Check solutions: Check if $\sin^2(\theta)$ can equal $\frac{1}{2}$ . This is true for $\theta = \frac{\pi}{4}$ or $\frac{3\pi}{4}$ (in the first and second quadrants where sine is positive).

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