Complete the proof of the identity by choosing the Rule that justifies (1-sin^2x)tan^2x=sin^2x To see a detailed description of a Rule , select the More Information B =sin^2x

To complete the proof of the identity \((1 - \sin^2 x) \tan^2 x = \sin^2 x\), we need to use trigonometric identities and algebraic manipulation.<br /><br />First, recall the Pythagorean identity:<br />\[ \sin^2 x + \cos^2 x = 1 \]<br /><br />From this identity, we can express \(1 - \sin^2 x\) as:<br />\[ 1 - \sin^2 x = \cos^2 x \]<br /><br />Next, recall the definition of \(\tan x\):<br />\[ \tan x = \frac{\sin x}{\cos x} \]<br /><br />Therefore, \(\tan^2 x\) can be written as:<br />\[ \tan^2 x = \left( \frac{\sin x}{\cos x} \right)^2 = \frac{\sin^2 x}{\cos^2 x} \]<br /><br />Now, substitute \(\cos^2 x\) for \(1 - \sin^2 x\) and \(\tan^2 x\) for \(\frac{\sin^2 x}{\cos^2 x}\) in the original equation:<br />\[ (1 - \sin^2 x) \tan^2 x = \cos^2 x \cdot \frac{\sin^2 x}{\cos^2 x} \]<br /><br />Simplify the right-hand side:<br />\[ \cos^2 x \cdot \frac{\sin^2 x}{\cos^2 x} = \sin^2 x \]<br /><br />Thus, we have shown that:<br />\[ (1 - \sin^2 x) \tan^2 x = \sin^2 x \]<br /><br />The rule that justifies this manipulation is the **Pythagorean Identity** and the **definition of tangent**.