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\frac{\tan\theta}{\sec\theta} = \sin\theta

Algebra Calculus Trigonometry Matrix

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

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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.

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}

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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

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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

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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)

\frac{\sin\left(\theta \right)}{1}

Divide the terms

\sin\left(\theta \right)

\sin\left(\theta \right)=\sin\left(\theta \right)

Solution

\textrm{true}

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