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The number of real solutions of the equation, sin1(i=1xi+1xi=1(x2)i)=π2cos1(i=1(x2)ii=1(x)i) lying in the interval (12,12) is,

Answer
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Hint: To solve this question first we will reduce the given equation into simplest form by using inverse trigonometric identity sin1x=π2cos1x, then we will expand the expansion and using sum of infinite G.P we will obtain the algebraic expression and then we will see how many root we obtain for that algebraic equation.

Complete step-by-step answer:
Now, we have sin1(i=1xi+1xi=1(x2)i)=π2cos1(i=1(x2)ii=1(x)i)
Let, f(x)=i=1xi+1xi=1(x2)i and g(x)=i=1(x2)ii=1(x)i
So, we can write sin1(i=1xi+1xi=1(x2)i)=π2cos1(i=1(x2)ii=1(x)i) as,
sin1f(x)=π2cos1g(x)
We know that, sin1x=π2cos1x,
So, re – writing sin1f(x)=π2cos1g(x) as,
sin1f(x)=sin1g(x)
Now, we know that sin1x is one - one function, so it is only true for sin1f(x)=sin1g(x),
 if f(x)=g(x) , that is
i=1xi+1xi=1(x2)i=i=1(x2)ii=1(x)i
Now, let us take f(x)=i=1xi+1xi=1(x2)i
We can write f(x) as, f(x)=i=1xixxi=1(x2)i
Or, f(x)=xi=1xixi=1(x2)i
Now, i=1xiis G.P whose first term is x and common ratio is x,
So, xi=1xi=x21x, as Sum of infinite G.P is a1r, whose first term is a and common ratio is r.
Similarly, i=1(x2)iis G.P whose first term is x2 and common ratio is x2,
So, xi=1(x2)i=x221(x2)=x22x.
So, f(x)=xi=1xixi=1(x2)i=x21xx22x
Now, in f(x)=i=1xi+1xi=1(x2)i and g(x)=i=1(x2)ii=1(x)i, there is only difference in sign,
So, g(x)=i=1(x2)ii=1(x)i=x21(x2)+x1(x)
Or on simplifying, we get
g(x)=i=1(x2)ii=1(x)i=x1+xx2+x
As, we have f(x)=g(x)
So, x21xx22x=x1+xx2+x
Now, we have to solve this algebraic expression,
Taking all expression on left side,
x21xx22xx1+x+x2+x=0
Taking, x common, we get
x(x1xx2x11+x+12+x)=0
So, x = 0 and (x1xx2x11+x+12+x)=0
Now, x1xx2x11+x+12+x=0x1xx2x=11+x12+x
Taking L.C.M, we get
2xx2x+x2(x2)(x1)=1x+2+x(x2)(x1)
On solving we get
x(x2)(x1)=1(x2)(x1)
On cross – multiplying, we get
x(x2)(x1)=(x2)(x1)
On opening brackets, we get
x3+2x2+5x2=0
Let, f(x)=x3+2x2+5x2
Now, at x=12 , we get
f(12)=(12)3+2(12)2+5(12)2=18+12522=338
at x=12 , we get
f(12)=(12)3+2(12)2+5(12)2=18+12+522=98
Now, f(12)f(12)<0, which means there exists one root in interval and one is x = 0.
So, we have two solutions for equation sin1(i=1xi+1xi=1(x2)i)=π2cos1(i=1(x2)ii=1(x)i)

Note: To, solve such question one must know the inverse trigonometric function properties such as sin1x=π2cos1x and also how to open summation and what is sum of infinite geometric expansion which is equals to a1r, whose first term is a and common ratio is r.