\frac{1+\tan \theta}{1+\cot \theta} = \tan \theta
Question
\frac{1+\tan\left(\theta \right)}{1+\cot\left(\theta \right)}=\tan\left(\theta \right)
Solve the equation
\text{Solve for }\theta
\theta \neq \left\{ \begin{array}{l}\frac{k\pi }{2}\\\frac{3\pi }{4}+k\pi \end{array}\right.,k \in \mathbb{Z}
Alternative Form
\theta \neq \left\{ \begin{array}{l}90^{\circ} k\\135^{\circ}+180^{\circ} k\end{array}\right.,k \in \mathbb{Z}
Evaluate
\frac{1+\tan\left(\theta \right)}{1+\cot\left(\theta \right)}=\tan\left(\theta \right)
Find the domain
More Steps Hide Steps
Evaluate
\left\{ \begin{array}{l}\theta \neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\\theta \neq k\pi ,k \in \mathbb{Z}\\1+\cot\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 \neq k\pi ,k \in \mathbb{Z}\\\theta \neq \frac{3\pi }{4}+k\pi ,k \in \mathbb{Z}\end{array}\right.
Calculate
\theta \neq \left\{ \begin{array}{l}\frac{k\pi }{2}\\\frac{3\pi }{4}+k\pi \end{array}\right.,k \in \mathbb{Z}
\frac{1+\tan\left(\theta \right)}{1+\cot\left(\theta \right)}=\tan\left(\theta \right),\theta \neq \left\{ \begin{array}{l}\frac{k\pi }{2}\\\frac{3\pi }{4}+k\pi \end{array}\right.,k \in \mathbb{Z}
Add or subtract both sides
\frac{1+\tan\left(\theta \right)}{1+\cot\left(\theta \right)}-\tan\left(\theta \right)=0
Simplify
More Steps Hide Steps
Evaluate
\frac{1+\tan\left(\theta \right)}{1+\cot\left(\theta \right)}-\tan\left(\theta \right)
Simplify
\frac{1+\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}}{1+\frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}}-\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}
Simplify
0
0=0
\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 \left\{ \begin{array}{l}\frac{k\pi }{2}\\\frac{3\pi }{4}+k\pi \end{array}\right.,k \in \mathbb{Z}
Solution
\theta \neq \left\{ \begin{array}{l}\frac{k\pi }{2}\\\frac{3\pi }{4}+k\pi \end{array}\right.,k \in \mathbb{Z}
Alternative Form
\theta \neq \left\{ \begin{array}{l}90^{\circ} k\\135^{\circ}+180^{\circ} k\end{array}\right.,k \in \mathbb{Z}
Verify the identity
Verify
\text{true}
Choose a side to work on
\frac{1+\tan\left(\theta \right)}{1+\cot\left(\theta \right)}=\tan\left(\theta \right)
Start working on the left-hand side
More Steps Hide Steps
Evaluate
\frac{1+\tan\left(\theta \right)}{1+\cot\left(\theta \right)}
Simplify
\frac{1+\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}}{1+\frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}}
Simplify
\frac{\left(\cos\left(\theta \right)+\sin\left(\theta \right)\right)\sin\left(\theta \right)}{\sin\left(\theta \right)\cos\left(\theta \right)+\cos^{2}\left(\theta \right)}
Simplify
\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}
Simplify
\tan\left(\theta \right)
\tan\left(\theta \right)=\tan\left(\theta \right)
Solution
\text{true}
Graph