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Question: How do you simplify \[{{\sin }^{2}}\left( \arcsin \left( \dfrac{2}{x} \right) \right)\]?...

How do you simplify sin⁡2(arcsin⁡(2x)){{\sin }^{2}}\left( \arcsin \left( \dfrac{2}{x} \right) \right)?

Explanation

Solution

For the given question we are given to find the solution of sin⁡2(arcsin⁡(2x)){{\sin }^{2}}\left( \arcsin \left( \dfrac{2}{x} \right) \right). By observing the problem we can see that there is arcsin⁡\arcsin inside the brackets. Now we have to consider the inside thing as a and then by solving the problem we will have the results.

Complete step by step solution:
Let us assume the given equation as ‘S’, we get
S=sin⁡2(arcsin⁡(2x))S={{\sin }^{2}}\left( \arcsin \left( \dfrac{2}{x} \right) \right)
Let us consider the above equation as equation (1), we get
S=sin⁡2(arcsin⁡(2x))......(1)S={{\sin }^{2}}\left( \arcsin \left( \dfrac{2}{x} \right) \right)......\left( 1 \right)
First of all we all have write the letter a and we all have to assume the given equation to get the solution
⇒S=sin⁡2(arcsin⁡(2x))\Rightarrow S={{\sin }^{2}}\left( \arcsin \left( \dfrac{2}{x} \right) \right)
Let us consider a is arcsin⁡(2x)\arcsin \left( \dfrac{2}{x} \right), we get
⇒a=arcsin⁡(2x)\Rightarrow a=\arcsin \left( \dfrac{2}{x} \right)
Then the sin a will 2 by x this sum will be not so easy to understand so we will derive it
⇒sin⁡a=2x\Rightarrow \sin a=\dfrac{2}{x}
And now we have to derive the expression of the value of the sin to derive the value by deriving it to the derivation of the derived quantity of the derived one
So, by squaring the above equation, we get the given problem, we get
⇒sin⁡2a=(2x)2\Rightarrow {{\sin }^{2}}a={{\left( \dfrac{2}{x} \right)}^{2}}
Let us consider the above equation as equation (2), we get
⇒sin⁡2a=(2x)2.................(2)\Rightarrow {{\sin }^{2}}a={{\left( \dfrac{2}{x} \right)}^{2}}.................\left( 2 \right)
After completing the total derivation we all get
⇒S=4x2\Rightarrow S=\dfrac{4}{{{x}^{2}}}
Let us consider the above equation as equation (3), we get
⇒S=4x2...................(3)\Rightarrow S=\dfrac{4}{{{x}^{2}}}...................\left( 3 \right)
Therefore, equation (3) is the solution for the given problem.

Note: This sum is based on the trigonometric values which is a difficult part of the whole mathematics. For solving any inverse trigonometric problem we have to consider the function inside the brackets as a variable and then solve the problem.