text: # How do you prove \frac { \tan x } { \sec x } + \frac { \cot x } { \csc x } = \sin x + \cos x?
  
  Dec 5, 2016
  
  See the explanation
  
  #### Explanation:
  
  $\frac{\textcolor{red}{\tan x}}{\textcolor{b l u e}{\sec x}} + \frac{\textcolor{g r e e n}{\cot x}}{\textcolor{m a \ge n t a}{\csc x}} = \sin x + \cos x$
  
  Rewrite $\textcolor{red}{\tan} x$ as $\frac{\textcolor{b r o w n}{\sin x}}{\textcolor{D a r k T u r q u o i s e}{\cos x}}$
  
  Rewrite $\textcolor{b l u e}{\sec} x$ as $\frac{1}{\textcolor{D a r k T u r q u o i s e}{\cos x}}$
  
  Rewrite $\textcolor{g r e e n}{\cot} x$ as $\frac{1}{\textcolor{red}{\tan x}}$
  
  Rewrite $\textcolor{m a \ge n t a}{\csc} x$ as $\frac{1}{\textcolor{b r o w n}{\sin x}}$
  
  $\frac{\frac{\textcolor{b r o w n}{\sin x}}{\textcolor{D a r k T u r q u o i s e}{\cos x}}}{\frac{1}{\textcolor{D a r k T u r q u o i s e}{\cos x}}} + \frac{\frac{1}{\textcolor{red}{\tan x}}}{\frac{1}{\textcolor{b r o w n}{\sin x}}} = \sin x + \cos x$
  
  $\frac{\textcolor{b r o w n}{\sin x} \cancel{\textcolor{D a r k T u r q u o i s e}{\cos x}}}{\cancel{\textcolor{D a r k T u r q u o i s e}{\cos x}}} + \frac{\textcolor{b r o w n}{\sin x}}{\textcolor{red}{\tan x}} = \sin x + \cos x$
  
  Rewrite $\textcolor{red}{\tan} x$ as $\frac{\textcolor{b r o w n}{\sin x}}{\textcolor{D a r k T u r q u o i s e}{\cos x}}$
  
  $\textcolor{b r o w n}{\sin} x + \frac{\textcolor{b r o w n}{\sin x}}{\frac{\textcolor{b r o w n}{\sin x}}{\textcolor{D a r k T u r q u o i s e}{\cos x}}} = \sin x + \cos x$
  
  $\textcolor{b r o w n}{\sin} x + \frac{\cancel{\textcolor{b r o w n}{\sin x}} \textcolor{D a r k T u r q u o i s e}{\cos x}}{\cancel{\textcolor{b r o w n}{\sin}}} x = \sin x + \cos x$
  
  $\textcolor{b r o w n}{\sin} x + \textcolor{D a r k T u r q u o i s e}{\cos} x = \sin x + \cos x$
explanation: The document contains a math question and a detailed step-by-step explanation of how to prove the given trigonometric identity.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # How do you prove \frac { \tan x } { \sec x } + \frac { \cot x } { \csc x } = \sin x + \cos x?
  
  Dec 5, 2016
  
  See the explanation
  
  #### Explanation:
  
  $\frac{\textcolor{red}{\tan x}}{\textcolor{b l u e}{\sec x}} + \frac{\textcolor{g r e e n}{\cot x}}{\textcolor{m a \ge n t a}{\csc x}} = \sin x + \cos x$
  
  Rewrite $\textcolor{red}{\tan} x$ as $\frac{\textcolor{b r o w n}{\sin x}}{\textcolor{D a r k T u r q u o i s e}{\cos x}}$
  
  Rewrite $\textcolor{b l u e}{\sec} x$ as $\frac{1}{\textcolor{D a r k T u r q u o i s e}{\cos x}}$
  
  Rewrite $\textcolor{g r e e n}{\cot} x$ as $\frac{1}{\textcolor{red}{\tan x}}$
  
  Rewrite $\textcolor{m a \ge n t a}{\csc} x$ as $\frac{1}{\textcolor{b r o w n}{\sin x}}$
  
  $\frac{\frac{\textcolor{b r o w n}{\sin x}}{\textcolor{D a r k T u r q u o i s e}{\cos x}}}{\frac{1}{\textcolor{D a r k T u r q u o i s e}{\cos x}}} + \frac{\frac{1}{\textcolor{red}{\tan x}}}{\frac{1}{\textcolor{b r o w n}{\sin x}}} = \sin x + \cos x$
  
  $\frac{\textcolor{b r o w n}{\sin x} \cancel{\textcolor{D a r k T u r q u o i s e}{\cos x}}}{\cancel{\textcolor{D a r k T u r q u o i s e}{\cos x}}} + \frac{\textcolor{b r o w n}{\sin x}}{\textcolor{red}{\tan x}} = \sin x + \cos x$
  
  Rewrite $\textcolor{red}{\tan} x$ as $\frac{\textcolor{b r o w n}{\sin x}}{\textcolor{D a r k T u r q u o i s e}{\cos x}}$
  
  $\textcolor{b r o w n}{\sin} x + \frac{\textcolor{b r o w n}{\sin x}}{\frac{\textcolor{b r o w n}{\sin x}}{\textcolor{D a r k T u r q u o i s e}{\cos x}}} = \sin x + \cos x$
  
  $\textcolor{b r o w n}{\sin} x + \frac{\cancel{\textcolor{b r o w n}{\sin x}} \textcolor{D a r k T u r q u o i s e}{\cos x}}{\cancel{\textcolor{b r o w n}{\sin}}} x = \sin x + \cos x$
  
  $\textcolor{b r o w n}{\sin} x + \textcolor{D a r k T u r q u o i s e}{\cos} x = \sin x + \cos x$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed step-by-step explanation of how to prove the given trigonometric identity.
  
  Label: question-and-answer