Simplify to a single trig function with no denominator. (tan^(2)theta)/(sec^(2)theta) Answer: theta

Express tan and sec: Express tan(theta) and sec(theta) in terms of sine and cosine. tan(theta) is the ratio of sin(theta) to cos(theta), so tan(theta) = sin(theta)/cos(theta). sec(theta) is the reciprocal of cos(theta), so sec(theta) = 1/cos(theta).Write tan^(2)theta: Write tan^(2)theta and sec^(2)theta using the expressions from Step 1. tan^(2)theta = (sin(theta)/cos(theta))^2 sec^(2)theta = (1/cos(theta))^2Substitute expressions: Substitute the expressions from Step 2 into the given expression. (tan^(2)theta)/(sec^(2)theta) = ((sin(theta)/cos(theta))^2) / ((1/cos(theta))^2)Simplify the expression: Simplify the expression by canceling out common terms. The cos(theta) in the denominator of tan^(2)theta and the cos(theta) in the numerator of sec^(2)theta will cancel out. This leaves us with (sin(theta))^2.Recognize sin^(2)theta: Recognize that (sin(theta))^2 is simply sin^(2)theta. So, the simplified expression is sin^(2)theta.

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Simplify to a single trig function with no denominator.

(tan^(2)theta)/(sec^(2)theta)
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theta

Simplify to a single trig function with no denominator. $\newline$ $\frac{\tan ^{2} \theta}{\sec ^{2} \theta}$ $\newline$ Answer:

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Algebra 2

Simplify expressions using trigonometric identities

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Q. Simplify to a single trig function with no denominator.

\newline

\frac{\tan ^{2} \theta}{\sec ^{2} \theta}

\newline

Answer:

Express tan and sec: Express $\tan(\theta)$ and $\sec(\theta)$ in terms of sine and cosine. $\newline$ $\tan(\theta)$ is the ratio of $\sin(\theta)$ to $\cos(\theta)$ , so $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ . $\newline$ $\sec(\theta)$ is the reciprocal of $\cos(\theta)$ , so $\sec(\theta) = \frac{1}{\cos(\theta)}$ .
Write $\tan^2\theta$ : Write $\tan^2\theta$ and $\sec^2\theta$ using the expressions from Step $1$ . $\newline$ $\tan^2\theta = \left(\frac{\sin(\theta)}{\cos(\theta)}\right)^2$ $\newline$ $\sec^2\theta = \left(\frac{1}{\cos(\theta)}\right)^2$
Substitute expressions: Substitute the expressions from Step $2$ into the given expression. $\newline$ $\frac{\tan^{2}\theta}{\sec^{2}\theta} = \frac{(\frac{\sin(\theta)}{\cos(\theta)})^2}{(\frac{1}{\cos(\theta)})^2}$
Simplify the expression: Simplify the expression by canceling out common terms. The $\cos(\theta)$ in the denominator of $\tan^{2}(\theta)$ and the $\cos(\theta)$ in the numerator of $\sec^{2}(\theta)$ will cancel out. This leaves us with $(\sin(\theta))^2$ .
Recognize $\sin^2\theta$ : Recognize that $(\sin(\theta))^2$ is simply $\sin^2\theta$ . So, the simplified expression is $\sin^2\theta$ .