\frac{\tan\theta}{\sec\theta} = \sin\theta
Question
\frac{\tan\left(\theta \right)}{\sec\left(\theta \right)}=\sin\left(\theta \right)
Solve the equation
\text{Solve for }\theta
\theta \neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}
Alternative Form
\theta \neq 90^{\circ}+180^{\circ} k,k \in \mathbb{Z}
Evaluate
\frac{\tan\left(\theta \right)}{\sec\left(\theta \right)}=\sin\left(\theta \right)
Find the domain
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Evaluate
\left\{ \begin{array}{l}\theta \neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\\sec\left(\theta \right)\neq 0\end{array}\right.
Rewrite the expression
\left\{ \begin{array}{l}\theta \neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\\theta \in \mathbb{R}\end{array}\right.
Calculate
\theta \neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}
\frac{\tan\left(\theta \right)}{\sec\left(\theta \right)}=\sin\left(\theta \right),\theta \neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}
Calculate
\sin\left(\theta \right)=\sin\left(\theta \right)
\text{The statement is true for any value of }\theta
\theta \in \mathbb{R}
Check if the solution is in the defined range
\theta \in \mathbb{R},\theta \neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}
Solution
\theta \neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}
Alternative Form
\theta \neq 90^{\circ}+180^{\circ} k,k \in \mathbb{Z}
Verify the identity
Verify
\text{true}
Choose a side to work on
\frac{\tan\left(\theta \right)}{\sec\left(\theta \right)}=\sin\left(\theta \right)
Start working on the left-hand side
\sin\left(\theta \right)=\sin\left(\theta \right)
Solution
\text{true}
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