Which expression is equivalent to [tex]$\tan (x-\pi)$[/tex]?
[tex]$\frac{\tan (x)-\tan (\pi)}{1-(\tan (x))(\tan (\pi))}$[/tex]
[tex]$\frac{\tan (x)-\tan (\pi)}{1+(\tan (x))(\tan (\pi))}$[/tex]
[tex]$\frac{\tan (x)+\tan (\pi)}{1-(\tan (x))(\tan (\pi))}$[/tex]
[tex]$\frac{\tan (x)+\tan (\pi)}{1+(\tan (x))(\tan (\pi))}$[/tex]
Answer (2)
Apply the tangent subtraction formula: tan ( x − π ) = 1 + t a n ( x ) t a n ( π ) t a n ( x ) − t a n ( π ) .
Substitute tan ( π ) = 0 into the expression.
Simplify the expression to get tan ( x − π ) = tan ( x ) .
The equivalent expression is 1 + ( t a n ( x )) ( t a n ( π )) t a n ( x ) − t a n ( π ) .
Explanation
Problem Analysis We are given the expression tan ( x − π ) and asked to find an equivalent expression from the given options. We will use the tangent subtraction formula to expand the given expression and simplify it using the value of tan ( π ) .
Applying Tangent Subtraction Formula The tangent subtraction formula is given by: tan ( a − b ) = 1 + tan ( a ) tan ( b ) tan ( a ) − tan ( b ) Applying this formula to tan ( x − π ) , we get: tan ( x − π ) = 1 + tan ( x ) tan ( π ) tan ( x ) − tan ( π )
Simplifying the Expression We know that tan ( π ) = 0 . Substituting this value into the expression, we have: tan ( x − π ) = 1 + tan ( x ) ⋅ 0 tan ( x ) − 0 = 1 + 0 tan ( x ) = 1 tan ( x ) = tan ( x ) Thus, tan ( x − π ) = tan ( x ) .
Comparing with Given Options Now, we need to find which of the given options is equivalent to tan ( x ) .
Option 1: 1 − ( t a n ( x )) ( t a n ( π )) t a n ( x ) − t a n ( π ) = 1 − ( t a n ( x )) ( 0 ) t a n ( x ) − 0 = 1 t a n ( x ) = tan ( x ) Option 2: 1 + ( t a n ( x )) ( t a n ( π )) t a n ( x ) − t a n ( π ) = 1 + ( t a n ( x )) ( 0 ) t a n ( x ) − 0 = 1 t a n ( x ) = tan ( x ) Option 3: 1 − ( t a n ( x )) ( t a n ( π )) t a n ( x ) + t a n ( π ) = 1 − ( t a n ( x )) ( 0 ) t a n ( x ) + 0 = 1 t a n ( x ) = tan ( x ) Option 4: 1 + ( t a n ( x )) ( t a n ( π )) t a n ( x ) + t a n ( π ) = 1 + ( t a n ( x )) ( 0 ) t a n ( x ) + 0 = 1 t a n ( x ) = tan ( x ) All the options are equivalent to tan ( x ) . However, the expression we derived using the tangent subtraction formula is 1 + ( t a n ( x )) ( t a n ( π )) t a n ( x ) − t a n ( π ) .
Final Answer Therefore, the expression equivalent to tan ( x − π ) is 1 + ( t a n ( x )) ( t a n ( π )) t a n ( x ) − t a n ( π ) .
Examples
Understanding trigonometric identities like the tangent subtraction formula is crucial in fields like physics and engineering. For instance, when analyzing the motion of a pendulum or the behavior of alternating current circuits, you often encounter expressions involving trigonometric functions. Simplifying these expressions using identities can make complex calculations more manageable and provide deeper insights into the system's behavior. This skill is also vital in signal processing, where trigonometric functions are used to represent and manipulate signals.
The expression equivalent to tan ( x − π ) is 1 + ( t a n ( x )) ( t a n ( π )) t a n ( x ) − t a n ( π ) , which simplifies to tan ( x ) because tan ( π ) = 0 .
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