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How do you simplify \[\dfrac{\sin \left( \theta +\pi \right)}{\cos \left( \theta -\pi \right)}\]?

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Last updated date: 20th Jun 2024
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Answer
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Hint: In this problem we have to simplify the given trigonometric equation. We should know that to solve these types of problems, we have to know the basic trigonometric formula, degree values, properties and identities. We can use trigonometric formulas to simplify and use trigonometric degree values to further simplify to get a simplified form of the given trigonometric expression.

Complete step by step answer:
We know that the given trigonometric expression to be simplified is,
\[\dfrac{\sin \left( \theta +\pi \right)}{\cos \left( \theta -\pi \right)}\] …….. (1)
We also know that the trigonometric formula can be used in this problem is,
\[\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B\] ……. (2)
\[\cos \left( A-B \right)=\cos A\cos B+\sin A\sin B\] ……. (3)
We can use the above two formulas in the trigonometric expression (1), we get
\[\Rightarrow \dfrac{\sin \theta \cos \pi +\cos \theta \sin \pi }{\cos \theta \cos \pi +\sin \theta \sin \pi }\]
Now we can simplify the above step using trigonometric degree value.
We know that \[\cos \pi =-1,\sin \pi =0\], we can substitute these values in the above step, we get
 \[\Rightarrow \dfrac{\sin \theta \left( -1 \right)+\cos \theta \left( 0 \right)}{\cos \theta \left( -1 \right)+\sin \theta \left( 0 \right)}\].
We can now multiply the terms both in numerator and denominator, we get
\[\Rightarrow \dfrac{-\sin \theta }{-\cos \theta }\]
We can now cancel the minus sign in both numerator and the denominator, we get
\[\Rightarrow \dfrac{\sin \theta }{\cos \theta }\]
We know that, \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\], we can apply this in the above step, we get
\[\Rightarrow \tan \theta \]

Therefore, the simplified form of \[\dfrac{\sin \left( \theta +\pi \right)}{\cos \left( \theta -\pi \right)}\] is \[\tan \theta \].

Note: Students make mistakes while substituting the correct formula, which should be concentrated. We should know that to solve these types of problems, we have to know the basic trigonometric formula, degree values, properties and identities. In this problem we have used several formulas which should be remembered as it is also helped in many problems in trigonometry.