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Question

If 3tan1 x+cot1 x=π, then x equals to

(a) 0        (b) 1        (c) 1        (d) 12


Solution

(b) Given that, 3tan1 x+cot1 x=π,

   2tan1 x+tan1 x+cot1 x=π              (i)         2tan1 x=ππ2           [ tan1 x+cot1 x=π2]   2tan1 x=π2   tan12x1x2=π2     [ 2tan1 x=tan12x1x2,  x (1, 1)]   2x1x2=tanπ2   2x1x2=10  1x2=0   x2=1  x=± 1  x=1

Hence, only x=1 satisfies the given equation.

Note Here, putting x=1 in the given equation, we get

3tan1(1)+cot1(1)=π   3tan1[tan(π4)]+cot1[cot(π4)]=π   3tan1(tantπ4)+cot1(cotπ4)=π   3tan1(tanπ4)+πcot1(cotπ4)=π     3.π4+ππ4=π           π+π=π  0π

Hence, x=1 does not satisfy the given equation.

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