Find cot x if sin x cot x csc x = sqrt2

2 Answer

  • Answer:

    The value of the cotangent of x i.e. cot x is:

                                  [tex]\cot x=\sqrt{2}[/tex]

    Step-by-step explanation:

    We are given a trignometric equation as follows:

    [tex]\sin x\cot x\csc x=\sqrt{2}[/tex]

    Now as we know that the cosecant function is the reciprocal of the sine function.

    i.e. we have:

    [tex]\csc x=\dfrac{1}{\sin x}[/tex]

    Hence, we solve our given expression in order to get the value of the cotangent function.

    [tex]\sin x\cot x\csc x=\sin x\csc x\cot x\\\\\sin x\cot x\csc x=\sin x\dfrac{1}{\sin x}\cot x\\\\\sin x\cot x\csc x=\cot x[/tex]


    [tex]\sin x\cot x\csc x=\sqrt{2}[/tex]

                            Hence, we get that:

                              [tex]\cot x=\sqrt{2}[/tex]

  • You can use the fact that multiplicative inverse of a trigonometric ratio is also a trigonometric ratio.

    The value of cot(x) for given condition is:

    [tex]cot(x) = \sqrt{2}[/tex]

    How are the trigonometric ratios related with multiplicative inverse?

    Multiplicative inverses are those numbers which when multiplied together give 1 as result which is called as identity element for multiplication.

    Thus, multiplicative inverse of x is 1/x since  [tex]x \times \dfrac{1}{x} = 1[/tex]

    The trigonometric ratios are related to their inverses as:

    [tex]\rm cosec(x) = \dfrac{1}{sin(x)}\\\\cot(x) = \dfrac{1}{tan(x)}\\\\sec(x) = \dfrac{1}{cos(x)}[/tex]

    How to find value of cot(x) for given case?

    Using aforesaid properties, we have:

    [tex]\rm sin(x)cot(x)csc(x) = \sqrt{x}\\\\cot(x) \times sin(x) \times csc(x) = \sqrt{2}\\\\\rm\\cot(x) \times sin(x) \times \dfrac{1}{sin(x)} = \sqrt{2} \text{\: (Since we have \:} csc(x) = \dfrac{1}{sin(x)})\\cot{x} = \sqrt{2}[/tex]


    The value of cot(x) for given condition is:

    [tex]cot(x) = \sqrt{2}[/tex]

    Learn more about sin and cosec here;