Question

Show that the following equation is impossible for any value of x.

$\sin \theta = \frac{1}{x^{2} } + {x}^{2}$
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Answer 1

Explanation :-

We are given, sin θ = x² + 1/x²

Let's solve it further,

→ sin θ = x² + 1/x²

→ sin θ = x² + 1/x² -2 + 2

[Adding and subtracting 2 will not change the equation]

→ sin θ = {x² + 1/x² - (2) (x) (1/x)} + 2 [x . 1/x = 1]

Now we see that the equation in curly braces is following an algebric identity a² + b² -2ab = (a-b)²

→ sin θ = ( x - 1/x )² + 2

It is clear that value of RHS is greater than 1 but we know that range of sin x function is [-1,1] that means value of sin x can't be greater than or smaller than 1 and -1 respectively.

Since LHS cannot be equal to RHS for any real values of x and θ, this situation is totally impossible.