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Question: How do you prove \(\tan \left( { - A} \right) = - \tan A?\)...

How do you prove tan⁡(−A)=−tan⁡A?\tan \left( { - A} \right) = - \tan A?

Explanation

Solution

In this question, we are going to prove that the value of the left hand side is equal to the value of the right hand side.
Here we are going to use the even and odd property to prove the trigonometric function.
Apply the quotient identities to the value of the left hand side and solving it we get the value of the right hand side.

Formula used: The Quotient identity is
tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{{\sin x}}{{\cos x}}
The even property is written as cos⁡(−x)=cos⁡x\cos \left( { - x} \right) = \cos x
The odd property is written as sin⁡(−x)=−sin⁡x\sin \left( { - x} \right) = - \sin x

Complete step-by-step solution:
In this question, we are going to prove that the value of one side is equal to the value of the other side.
First, write the given equation tan⁡(−A)=−tan⁡A\tan \left( { - A} \right) = - \tan A
Take the left hand side: tan⁡(−A)\tan \left( { - A} \right)
Use the quotient identity to the given equation
tan⁡(−A)=sin⁡(−A)cos⁡(−A)\tan \left( { - A} \right) = \dfrac{{\sin \left( { - A} \right)}}{{\cos \left( { - A} \right)}}
Now use the odd property sin⁡(−A)=−sin⁡A\sin \left( { - A} \right) = - \sin A to the above equation we get
⇒tan⁡(−A)=−sin⁡(A)cos⁡(−A)\Rightarrow \tan \left( { - A} \right) = \dfrac{{ - \sin \left( A \right)}}{{\cos \left( { - A} \right)}}
Next use the even property cos⁡(−A)=cos⁡A\cos \left( { - A} \right) = \cos A to the above equation we get
⇒tan⁡(−A)=−sin⁡(A)cos⁡(A)\Rightarrow \tan \left( { - A} \right) = \dfrac{{ - \sin \left( A \right)}}{{\cos \left( A \right)}}
On rewriting we get
⇒tan⁡(−A)=−sin⁡(A)cos⁡(A)\Rightarrow \tan \left( { - A} \right) = - \dfrac{{\sin \left( A \right)}}{{\cos \left( A \right)}}
By using quotient identity we can write the above equation as
⇒tan⁡(−A)=−tan⁡A\Rightarrow \tan \left( { - A} \right) = - \tan A
Thus the value of the left hand side of the equation is equal to the value of the right hand side of the equation.
Thus we have proved the result.

Note: We can prove the identity by reducing each side separately. We do not know if the two sides are equal. At the end we ended up with the same thing, so we can conclude that it is valid.
There are several options we can use when proving a trigonometric identity.
Often, one of the steps for Proving identities to change each term into their sine and cosine equivalents
Use Pythagorean Theorem and other fundamental identities.
When working with identities where there are fractions combine using algebraic techniques for adding expressions with unlike denominators.
If possible factor trigonometric expressions.