In this article, we will explore the solution of the trigonometric equation 2sin^{-1}xsin^{-1}2xfrac{pi}{2}. This problem involves the application of inverse sine functions, sine identities, and algebraic manipulation. Let's break it down step by step:

Solution Part 1

To begin, we can isolate one of the inverse sine terms. Start with the given equation:

2sin^{-1}x sin^{-1}2x frac{pi}{2}

Isolate sin^{-1}2x:

sin^{-1}2x frac{pi}{2} - 2sin^{-1}x

Now, apply the sine function to both sides:

2x sinleft(frac{pi}{2} - 2sin^{-1}xright)

Use the identity sinleft(frac{pi}{2} - thetaright) costheta to simplify:

2x cosleft(2sin^{-1}xright)

Apply the double angle formula for cosine, cos2theta 1 - 2sin^2theta, where theta sin^{-1}x and hence:

cosleft(2sin^{-1}xright) 1 - 2sin^2left(sin^{-1}xright) 1 - 2x^2

Substitute this into the previous equation:

2x 1 - 2x^2

Rearrange the equation to form a quadratic equation:

2x^2 - 2x - 1 0

Solution Part 2

Use the quadratic formula to solve for x. The quadratic formula is given by:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Where a 2, b -2, and c -1. Substitute these values into the formula:

x frac{-(-2) pm sqrt{(-2)^2 - 4 cdot 2 cdot (-1)}}{2 cdot 2} frac{2 pm sqrt{4 8}}{4} frac{2 pm sqrt{12}}{4} frac{2 pm 2sqrt{3}}{4} frac{1 pm sqrt{3}}{2}

We now have two potential solutions:

x frac{1 sqrt{3}}{2} x frac{1 - sqrt{3}}{2}

Validation of Solutions

We need to check which of these solutions are valid within the range of sin^{-1}, which is [-1, 1].

x frac{1 sqrt{3}}{2} approx 0.366 is within the range [-1, 1], so it is a valid solution. x frac{1 - sqrt{3}}{2} approx -1.366 is not within the range, so it is not a valid solution.

Final Solution

The only valid solution is:

boxed{frac{1 - sqrt{3}}{2}}

Alternative Solution Method

Let sin^{-1}x alpha implies x sin alpha.

Let sin^{-1}2x beta implies 2x sin beta.

Consider the following diagram for other values:

Given: 2sin^{-1}x sin^{-1}2x frac{pi}{2} implies 2alpha beta frac{pi}{2}.

Taking the cosine of both sides:

cos2alpha beta cosfrac{pi}{2} 0

Implies cos2alpha cosbeta - sin2alpha sinbeta 0

Implies tan2alpha cotbeta

Implies frac{2tanalpha}{1 - tan^2alpha} frac{1}{tanbeta}

Implies frac{2x}{sqrt{1-x^2}(1 - frac{x^2}{1-x^2})} frac{sqrt{1-4x^2}}{2x}

Implies frac{4x^2}{1 - 2x^2} sqrt{frac{1-4x^2}{1-x^2}}

Squaring both sides:

Implies frac{16x^4}{1 - 2x^2} frac{1-4x^2}{1-x^2}

Implies 16x^4 (1 - x^2) (1-4x^2)(1 - 2x^2)

Implies 16x^4 - 16x^6 1 - 4x^2 - 4x^4 8x^4

Implies 4x^4 - 8x^2 - 1 0

Implies x^2 frac{8 pm sqrt{64 - 16}}{8} frac{8 pm 4sqrt{3}}{8}

Implies x^2 frac{2 pm sqrt{3}}{2}

Implies x pm sqrt{frac{2 pm sqrt{3}}{2}}

Validation of Solutions (Alternative Method)

Check the 4 possibilities:

x sqrt{frac{2 sqrt{3}}{2}} approx 1.366 is not within the range, so it is not a valid solution. x -sqrt{frac{2 sqrt{3}}{2}} approx -1.366 is not within the range, so it is not a valid solution. x -sqrt{frac{2 - sqrt{3}}{2}} frac{sqrt{3}-1}{2} is approximately 0.37437, and this value satisfies the equation, making it a valid solution. x sqrt{frac{2 - sqrt{3}}{2}} approx 0.37437, but this is also a valid solution as it is positive and within the range.

After validation, the valid solutions are:

x -sqrt{frac{2 - sqrt{3}}{2}} frac{sqrt{3}-1}{2} and other valid ones within the range.

The valid solution is:

boxed{frac{sqrt{3}-1}{2}}