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Inverse Trigonometric Functions

Question

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Solve $tan to the power of negative 1 end exponent 2 x plus tan to the power of negative 1 end exponent 3 x equals straight pi over 4.$

Solution

tan to the power of negative 1 end exponent 2 x plus tan to the power of negative 1 end exponent 3 x equals straight pi over 4 space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space tan to the power of negative 1 end exponent open square brackets fraction numerator 2 x plus 3 x over denominator 1 minus 2 x.3 x end fraction close square brackets rightwards double arrow space space space space fraction numerator 5 x over denominator 1 minus 6 x squared end fraction equals tan straight pi over 4 space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space fraction numerator 5 x over denominator 1 minus 6 x squared end fraction equals 1 space rightwards double arrow space space space 5 x equals 1 minus 6 x squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space 6 x squared plus 5 x minus 1 equals 0 space rightwards double arrow space space space left parenthesis 6 x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis equals 0 space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space x equals 1 over 6. negative 1 space space space

Now x = – 1 does not satisfy the given equation as the L.H.S. of the equation becomes negative

therefore space space space straight x equals 1 over 6

is the only solution of the given equation.

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