Questions of Multiple Angles of Trigonometry

1. Prove that:
(a) \tan4\text{A} = \displaystyle{4\tan\text{A} – 4\tan^3\text{A}\over{1 – 6\tan^2\text{A} + \tan^4\text{A}}}




(b) 



(c) sin 5A = 16sin5A – 20sin3A + 5sinAsin^5\text{A} – 20\sin^3\text{A} + 5\sin\text{A}



(d) \tan5\text{A} = \displaystyle{4\tan\text{A} – 10\tan^3\text{A} + \tan^5\text{A}\over{1 – 10\tan^2\text{A} + 5\tan^4\text{A}}} 



(e) sin6A = sin2A(16sin4A – 16sin2A + 3)



(f) cos6A = 32cos6A – 48cos4A + 18cos2A – 1



2. Prove that:
(a) \displaystyle{2\sin\text{x}\over{\cos\text{3x}}} + \displaystyle{2\sin3\text{x}\over{\cos\text{9x}}} + \displaystyle{2\sin9\text{x}\over{\cos27\text{x}}} = \tan27\text{x} – \tan\text{x} 




(b) \cos\displaystyle{13\pi\over{45}} + \cos\displaystyle{17\pi\over{45}} + \cos\displaystyle{43\pi\over{45}} = 0 



3. Prove that:
(a) \tan^620° – 33\tan^420° + 27\tan^220° = 3 



(b) 64\tan^620° – 96\sin^420° + 36\sin^220° = 3 



(c) tan9°– tan27°tan63°+tan81°=4




(d) 1+2sin20°.tan40°.tan60°.tan80°=8cos³20°    




(e)  \sin^6\displaystyle{\pi^c\over{8}} + \sin^6\displaystyle{3\pi^c\over{8}} + \sin^6\displaystyle{5\pi^c\over{8}} + \sin^6\displaystyle{7\pi^c\over{8}} = \displaystyle{5\over{4}} 




(f) \tan50° – \displaystyle{4\sin50°\over{\sqrt{3}}} = –\displaystyle{1\over{3}}\tan50° – \displaystyle{4\sin50°\over{\sqrt{3}}} = –\displaystyle{1\over{\sqrt{3}}} 





4.(a) \pi^If A + B + C = \pi^c and cosA = cosB.cosC, prove that:
(i) tanA = tanB + tanC



(ii) \cot\text{B}.\cot\text{C} = \displaystyle{1\over{2}} 


(b) \text{A + B + C} = \displaystyle{\pi^c\πIf \text{A + B + C} = \displaystyle{\pi^c\over{2}} and sinA.sinB = sinC, prove that cotA. cotB = 2.



(c) If A + B = 225°, prove that: \displaystyle{\cot\text{A}\over{1 + \cot\text{A}}}.\displaystyle{\cot\text{B}\over{1 + \cot\text{B}}} = \displaystyle{1\over{2}} 



(d)If \displaystyle{1 + \sin2\text{A}\over{1 + \cos2\text{A}}} = \displaystyle{1 + \cos2\text{B}\over{1 + \sin2\text{B}}} , prove that (\text{A + B}) = \displaystyle{\pi^c\over{4}}






(e) If \cos\text{A} = \displaystyle{3\over{5}} and \sin\text{B} = \displaystyle{12\over{13}} , prove that:
(i) \sin^2\bigg(\displaystyle{\text{A – B}\over{2}}\bigg) = \displaystyle{1\over{16}} 


(ii) \cos\bigg(\displaystyle{\text{A – B}\over{2}}\bigg) = \displaystyle{8\over{\sqrt{65}}}




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