Respuesta :
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]
As:
[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]
Thus,
The value of cot(x) for given condition is:
[tex]cot(x) = \sqrt{2}[/tex]
Learn more about sin and cosec here;
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