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The value of \(\rm \frac{{\cos A - \cos 5A - \cos 9A + \cos 13A}}{{\sin A - \sin 5A + \sin 9A - \sin 13A}}\) is
1. cot 2A
2. tan 2A
3. cot 4A
4. tan 4A

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Correct Answer - Option 4 : tan 4A

Formula Used:

cos A  + cos B = 2 cos (A + B)/2 × cos (A – B)/2

cos A  - cos B = - 2 sin (A + B)/2 × sin (A – B)/2

sin A – sin B = 2 cos (A + B)/2 × sin (A – B)/2

sin (- A) = - sin (A)

Calculation:

⇒ \(\rm \frac{{\cos A - \cos 5A - \cos 9A + \cos 13A}}{{\sin A - \sin 5A + \sin 9A - \sin 13A}}\)

⇒ \(\frac{{(\cos A + \cos 13A) - (\cos 5A + \cos 9A)\;}}{{(\sin A - \sin 13A) + ( - \sin 5A + \sin 9A)\;}}\)

⇒ \(\frac{{2\cos \frac{{14A}}{2}\cos \frac{{12A}}{2}\; - 2\cos \frac{{14A}}{2}\cos \frac{{4A}}{2}}}{{2\cos \frac{{14A}}{2}\sin \left( { - \frac{{12A}}{2}} \right) - 2\cos \frac{{14A}}{2}\sin \frac{{4A}}{2}}}\)

⇒ \(\frac{{2\cos \frac{{14A}}{2}[\cos 6A - \cos 2A]}}{{2\cos \frac{{14A}}{2}[ - \sin 6A + \sin 2A]}}\)

⇒ \(\frac{{[\cos 6A - \cos 2A]}}{{[\sin 2A - \sin 6A]}}\)

⇒ \(\frac{{ - 2\sin \frac{{8A}}{2}\sin \frac{{4A}}{2}}}{{ - 2\cos \frac{{8A}}{2}\sin \frac{{4A}}{2}}}\)

⇒ \(\frac{{\sin 4A}}{{\cos 4A}}\)

⇒ tan 4A

∴ The value of \(\rm \frac{{\cos A - \cos 5A - \cos 9A + \cos 13A}}{{\sin A - \sin 5A + \sin 9A - \sin 13A}}\) is tan 4A.

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