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Question
\frac{\tan\left(\theta \right)}{\sec\left(\theta \right)}=\sin\left(\theta \right)
Solve the equation
\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
More Steps
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.
Calculate
\left\{ \begin{array}{l}\theta \neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\\theta \in \mathbb{R}\end{array}\right.
Find the intersection
\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}
Rewrite the expression
\frac{\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}}{\frac{1}{\cos\left(\theta \right)}}=\sin\left(\theta \right)
Simplify the expression
\frac{\sin\left(\theta \right)\cos\left(\theta \right)}{\cos\left(\theta \right)}=\sin\left(\theta \right)
Simplify the expression
\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}
Show Solutions
Verify the identity
\textrm{true}
Evaluate
\frac{\tan\left(\theta \right)}{\sec\left(\theta \right)}=\sin\left(\theta \right)
Start working on the left-hand side
More Steps
Evaluate
\frac{\tan\left(\theta \right)}{\sec\left(\theta \right)}
\text{Use }\tan t = \frac{\sin t}{\cos t}\text{ to transform the expression}
\frac{\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}}{\sec\left(\theta \right)}
Multiply by the reciprocal
\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}\times \frac{1}{\sec\left(\theta \right)}
Multiply the terms
\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)\sec\left(\theta \right)}
Transform the expression
More Steps
Evaluate
\cos\left(\theta \right)\sec\left(\theta \right)
\text{Use }\sec t = \frac{1}{\cos t}\text{ to transform the expression}
\cos\left(\theta \right)\times \frac{1}{\cos\left(\theta \right)}
\text{Cancel out the common factor }\cos\left(\theta \right)
1\times 1
Multiply the terms
1
\frac{\sin\left(\theta \right)}{1}
Divide the terms
\sin\left(\theta \right)
\sin\left(\theta \right)=\sin\left(\theta \right)
Solution
\textrm{true}
Show Solutions
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