Home > Math Solver

Math Solver

Topics

Pre Algebra

Fractions

Decimals

Exponents

Radicals

Ratios

Integer

Percent

Algebra

Equations

Expressions

Inequalities

Functions

Sequences

Logarithmetic

Complex Numbers

Matrices

Pre Calculus

Equations

Inequalities

Functions

Calculus

Derivatives

Integrals

Limits

Series

Trigonometry

Equations

Inequalities

Functions

Simplify

Proof

Geometry

Statistics and Probability

Matrix

\frac{1+\tan \theta}{1+\cot \theta} = \tan \theta

Algebra Calculus Trigonometry Matrix

Question

\frac{1+\tan\left(\theta \right)}{1+\cot\left(\theta \right)}=\tan\left(\theta \right)

Solve the equation

\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.

Calculate

\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.

Find the intersection

\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}

Rewrite the expression

\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)}

Calculate

\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}=\frac{\sin\left(\theta \right)}{\cos\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 \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}

Show Solutions Hide Solutions

Verify the identity

\textrm{true}

Evaluate

\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)}

Transform the expression

More Steps Hide Steps

Evaluate

1+\tan\left(\theta \right)

\text{Use }\tan t = \frac{\sin t}{\cos t}\text{ to transform the expression}

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

Reduce fractions to a common denominator

\frac{\cos\left(\theta \right)}{\cos\left(\theta \right)}+\frac{\sin\left(\theta \right)}{\cos\left(\theta \right)}

Write all numerators above the common denominator

\frac{\cos\left(\theta \right)+\sin\left(\theta \right)}{\cos\left(\theta \right)}

\frac{\frac{\cos\left(\theta \right)+\sin\left(\theta \right)}{\cos\left(\theta \right)}}{1+\cot\left(\theta \right)}

Transform the expression

More Steps Hide Steps

Evaluate

1+\cot\left(\theta \right)

\text{Use }\cot t = \frac{\cos t}{\sin t}\text{ to transform the expression}

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

Reduce fractions to a common denominator

\frac{\sin\left(\theta \right)}{\sin\left(\theta \right)}+\frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}

Write all numerators above the common denominator

\frac{\sin\left(\theta \right)+\cos\left(\theta \right)}{\sin\left(\theta \right)}

\frac{\frac{\cos\left(\theta \right)+\sin\left(\theta \right)}{\cos\left(\theta \right)}}{\frac{\sin\left(\theta \right)+\cos\left(\theta \right)}{\sin\left(\theta \right)}}

Multiply by the reciprocal

\frac{\cos\left(\theta \right)+\sin\left(\theta \right)}{\cos\left(\theta \right)}\times \frac{\sin\left(\theta \right)}{\sin\left(\theta \right)+\cos\left(\theta \right)}

\text{Cancel out the common factor }\cos\left(\theta \right)+\sin\left(\theta \right)

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

Multiply the terms

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

\text{Use }\frac{\sin(t)}{\cos(t)}=\tan(t)\text{ to transform the expression}

\tan\left(\theta \right)

\tan\left(\theta \right)=\tan\left(\theta \right)

Solution

\textrm{true}

Show Solutions Hide Solutions

Graph

Still have questions?
Ask UpStudy online

24/7 expert live tutors
Unlimited numbers of questions
Step-by-step explanations

Math Solver

Still have questions? Ask UpStudy online

Still have questions?
Ask UpStudy online