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Simplify to a single trig function with no denominator.

tan^(2)theta*csc^(2)theta
Answer:

theta

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

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Simplify expressions using trigonometric identities

Full solution

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

\newline

\tan ^{2} \theta \cdot \csc ^{2} \theta

\newline

Answer:

Express $\tan$ and $\csc$ : Express $\tan(\theta)$ and $\csc(\theta)$ in terms of $\sin(\theta)$ and $\cos(\theta)$ . $\newline$ $\tan(\theta)$ is $\frac{\sin(\theta)}{\cos(\theta)}$ and $\csc(\theta)$ is $\frac{1}{\sin(\theta)}$ .
Write $\tan^2$ and $\csc^2$ : Write $\tan^2\theta$ and $\csc^2\theta$ using the expressions from Step $1$ . $\newline$ $\tan^2\theta$ becomes $(\sin(\theta)/\cos(\theta))^2$ and $\csc^2\theta$ becomes $(1/\sin(\theta))^2$ .
Multiply the expressions: Multiply the expressions from Step $2$ . $\newline$ $(\frac{\sin(\theta)}{\cos(\theta)})^2 \times (\frac{1}{\sin(\theta)})^2$ simplifies to $(\frac{\sin^2(\theta)}{\cos^2(\theta)}) \times (\frac{1}{\sin^2(\theta)})$ .
Simplify the expression: Simplify the expression by canceling out $\sin^2(\theta)$ in the numerator and denominator. $\frac{\sin^2(\theta)}{\cos^2(\theta)} \times \frac{1}{\sin^2(\theta)}$ simplifies to $\frac{1}{\cos^2(\theta)}$ .
Recognize the definition: Recognize that $\frac{1}{\cos^2(\theta)}$ is the definition of $\sec^2(\theta)$ . Therefore, the simplified expression is $\sec^2(\theta)$ .

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