Question

Prove that if x and y are not odd multiple of \[\dfrac{\pi }{2}\] then \[\tan x=\tan y\Rightarrow x=n\pi +y\], where \[n\in z\].

Answer 1

Hint: Write the given relation using the conversion: - \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\] and cross multiply the terms of the fraction. Now, take all the terms to the L.H.S and use the identity \[\sin a\cos b-\cos a\sin b=\sin \left( a-b \right)\] to form a trigonometric equation. Use the formula \[\sin \theta =0\Rightarrow \theta =n\pi ,n\in z\], to get the required proof.

Complete step-by-step solution
Here, we have been provided with the equation, \[\tan x=\tan y\] and we have to show, \[x=n\pi +y,n\in z\].
Now, using the relation: - \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\], we get,
\[\Rightarrow \dfrac{\sin x}{\cos x}=\dfrac{\sin y}{\cos y}\]
By cross – multiplication we get,
\[\Rightarrow \sin x\cos y=\sin y\cos x\]
Taking all the terms to the L.H.S, we get,
\[\Rightarrow \sin x\cos y-\cos x\sin y=0\]
Using the identity: - \[\sin a\cos b-\cos a\sin b=\sin \left( a-b \right)\], we get,
\[\Rightarrow \sin \left( x-y \right)=0\] - (1)
Now, we know that if \[\sin \theta =0\], then its general solution is given by the relation: - \[\theta =n\pi \], where \[n\in z\]. Here ‘n’ cannot be any fractional number but it should always be an integer whose set is denoted by z. So, the general solution of equation (1) can be given as: -
\[\Rightarrow x-y=n\pi \], where \[n\in z\].
\[\Rightarrow x=n\pi +y\], where \[n\in z\].
Hence proved
Now, here x and y cannot be an odd multiple of \[\dfrac{\pi }{2}\] because tangent of any angle is undefined if the angle is an odd multiple of \[\dfrac{\pi }{2}\]. This is because \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\] and at angles like \[\pm \dfrac{\pi }{2},\pm \dfrac{3\pi }{2},....\] and so on, the value of \[\cos \theta =0\].

Note: One may note that here we have proved \[x=n\pi +y\]. Here, remember that ‘n’ should always be an integer so do not forget to mention it in the last step of the answer otherwise the answer will be considered as incomplete. Remember that \[\tan \theta \] and \[\sec \theta \] are undefined at odd multiples of \[\dfrac{\pi }{2}\] and \[\cot \theta \] and \[\csc \theta \] are undefined at integral multiples of \[\pi \]. Remember the formulas for the general solution of all the trigonometric functions.