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2014-08-03T22:09:16+05:30

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Take RHS
4 cot 2A cosec 2A = 4 cos 2A / (sin 2A * sin 2A)
cos 2A = cos² A - sin² A
sin 2A = 2 sin A cos A
RHS = 4 ( cos² A - sin² A )  / 4 sin² A  cos²  A
       = cos² A  -  sin² A
         --------------------                =   
         cos²  A  sin² A

=  1/ sin² A  - 1/cos² A     NOW multiply  by cos² A  and divide by cos² A
=  cosec² A - sec² A  =  (cot² A + 1 ) - (tan² A + 1 )
 = cot² A - tan² A

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2014-08-04T02:41:39+05:30
a=x\\\\cot^2x-tan^2x=4cot2xcosec2x\\\\L=(cotx-tanx)(cotx+tanx)=\left(\frac{cosx}{sinx}-\frac{sinx}{cosx}\right)\left(\frac{cosx}{sinx}+\frac{sinx}{cosx}\right)\\\\=\left(\frac{cos^2x}{sinxcosx}-\frac{sin^2x}{sinxcosx}\right)\left(\frac{cos^2x}{sinxcosx}+\frac{sin^2x}{sinxcosx}\right)\\\\=\frac{cos^2x-sin^2x}{sinxcosx}\times\frac{cos^2x+sin^2x}{sinxcosx}=\frac{cos2x}{sinxcosx}\times\frac{1}{sinxcosx}

=\frac{cos2x}{\frac{1}{2}\times2sinxcosx}\times\frac{1}{\frac{1}{2}\times2sinxcosx}=\frac{cos2x}{\frac{1}{2}sin2x}\times\frac{1}{\frac{1}{2}sin2x}=2\times\frac{cos2x}{sin2x}\times2\times\frac{1}{sin2x}\\\\=4ctg2xcosec2x=R

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Used:\\\\a^2-b^2=(a-b)(a+b)\\\\tanx=\frac{sinx}{cosx}\\\\cotx=\frac{cosx}{sinx}\\\\sin^2x+cos^2x=1\\\\cos2x=cos^2x-sin^2x\\\\sin2x=2sinxcosx\\\\cosecx=\frac{1}{sinx}
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