EXAMPLE 30 Prove that tan = 3 tan-1...
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EXAMPLE 30 Prove that tan = 3 tan-1

11th - 12th Class
Maths
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soution Putting ( x=a ) tan ( theta, ) we get [ begin{aligned} tan ^{-1}left(frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}right) &=tan ^{-1}left(frac{3 a^{3} tan theta-a^{3} tan ^{3} theta}{a^{3}-3 a^{3} tan ^{2} theta}right) &=tan ^{-1}left(frac{3 tan theta-tan ^{3} theta}{1-3 tan ^{2} theta}right)=tan ^{-1}(tan 3 theta) &=3 theta=3 tan ^{-1} frac{x}{a} text { Hence, } tan ^{-1}left(frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}right) &=3 tan ^{-1} frac{x}{a} end{aligned} ]
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