a. (sin^3Theta )/(cos^2)Theta (1 point) Which expression is equivalent to tan^2Theta -sin^2Theta

To solve this problem, we need to simplify the given expression \( \tan^2 \theta - \sin^2 \theta \) and determine which of the provided options is equivalent.<br /><br />### Step 1: Recall the definitions of trigonometric functions<br />- The tangent function is defined as:<br /> \[<br /> \tan \theta = \frac{\sin \theta}{\cos \theta}<br /> \]<br /> Therefore,<br /> \[<br /> \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}.<br /> \]<br /><br />### Step 2: Substitute \( \tan^2 \theta \) into the expression<br />The given expression is:<br />\[<br />\tan^2 \theta - \sin^2 \theta.<br />\]<br />Substituting \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \), we get:<br />\[<br />\tan^2 \theta - \sin^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} - \sin^2 \theta.<br />\]<br /><br />### Step 3: Combine terms under a common denominator<br />To combine the terms, rewrite \( \sin^2 \theta \) with a denominator of \( \cos^2 \theta \):<br />\[<br />\frac{\sin^2 \theta}{\cos^2 \theta} - \sin^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} - \frac{\sin^2 \theta \cos^2 \theta}{\cos^2 \theta}.<br />\]<br />Now, combine the fractions:<br />\[<br />\frac{\sin^2 \theta}{\cos^2 \theta} - \frac{\sin^2 \theta \cos^2 \theta}{\cos^2 \theta} = \frac{\sin^2 \theta - \sin^2 \theta \cos^2 \theta}{\cos^2 \theta}.<br />\]<br /><br />### Step 4: Factor out \( \sin^2 \theta \) in the numerator<br />In the numerator \( \sin^2 \theta - \sin^2 \theta \cos^2 \theta \), factor out \( \sin^2 \theta \):<br />\[<br />\sin^2 \theta - \sin^2 \theta \cos^2 \theta = \sin^2 \theta (1 - \cos^2 \theta).<br />\]<br /><br />### Step 5: Simplify using the Pythagorean identity<br />From the Pythagorean identity, \( 1 - \cos^2 \theta = \sin^2 \theta \). Substituting this, we get:<br />\[<br />\sin^2 \theta (1 - \cos^2 \theta) = \sin^2 \theta \cdot \sin^2 \theta = \sin^4 \theta.<br />\]<br /><br />Thus, the numerator becomes \( \sin^4 \theta \), and the entire expression is:<br />\[<br />\frac{\sin^4 \theta}{\cos^2 \theta}.<br />\]<br /><br />### Step 6: Match the result with the options<br />The simplified expression is:<br />\[<br />\frac{\sin^4 \theta}{\cos^2 \theta}.<br />\]<br />This matches **Option A**:<br />\[<br />\frac{\sin^3 \theta}{\cos^2 \theta}.<br />\]