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the number of distinct real roos of the equation tan22x+2tan2xtan3x-1=0 in the interval(0,90)
6simply put tan^x=ywe also know that tan2x=2tanx/(1-tan^2x) and tan3x= (3tanx-tan^3x)/(1-3tan^2x)so putting al these in the given equation, and removing the denominators with simmplification gives:7y^3-35y^2+21y-1=0 (Note that 0let f(y)= 7y^3-35y^2+21y-1now we see that f(0)*f(1/3)also, f(1/3)*f(1)and since f(1) so the given equation has 3 real roots
simply put tan^2x=ywe also know that tan2x=2tanx/(1-tan^2x) and tan3x= (3tanx-tan^3x)/(1-3tan^2x)so putting all these in the given equation, and removing the denominators with simmplification gives:7y^3-35y^2+21y-1=0 (Note that 0let f(y)= 7y^3-35y^2+21y-1now we see that f(0)*f(1/3)also, f(1/3)*f(1)and since f(1) is negative, one root also lies between 1 and infinityso the given equation has 3 real roots
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