Does this property holds for inverse too?
cos − 1 ( x ) sin − 1 ( x ) = tan − 1 ( x )
Determine the number of real solution of x which satisfy the equation above.
Original problem.
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3 solutions
Well, which is the difficult way?
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Sorry, I wasn't suggesting a difficult way, just to say that graphing is a need-no-thinking easy way.
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As this problem falls under Algebra section, I was just wondering how to approach it algebraically(one way I found was to find Maclaurin series of both LHS and RHS) but that indeed involves some complexity. By the way, while plotting the graph without any application, how do we find the second point of intersection ( 0 being the obvious 1 st one)?
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@Vighnesh Raut – I plotted the graph and approximated for the second point with an Excel spreadsheet.
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@Chew-Seong Cheong – Oh, like Numerical Analysis right? Sorry for asking so many questions. Just a bit curious about it.
The solution is
x = 0 a n d x = 4 π . I f x = 0 , c o s − 1 0 S i n − 1 0 = π / 2 0 = 0 = T a n − 1 0 .
I f x = 4 π . , c o s − 1 4 π S i n − 1 4 π = 1 = T a n − 1 4 π .
@Niranjan Khanderia I've converted it into a solution.
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Guess and check's the only way?
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Nope. Plotting the graph as shown by @Chew-Seong Cheong is a much better way.
tan^-1(x) is the inverse tan function, so tan^-1(pi/4) definitely does not equal 1.
sin − 1 4 π ≈ 0 . 9 0 3 3 3 9 1 1 1 , cos − 1 4 π ≈ 0 . 6 6 7 4 5 7 2 1 6 and tan − 1 4 π ≈ 0 . 6 6 5 7 7 3 7 5 . ⟹ cos − 1 4 π sin − 1 4 π = tan − 1 4 π .
Plot both the graphs. One will get two intersections. These two intersections are the solutions of the given question
Wothout using graphing app,how will you draw cos − 1 x sin − 1 x ?
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It is possible to draw by just substituting some values of 'x' and drawing an approximate graph.
As suggested by @Saarthak Marathe , an easy way to solve this is by plotting. Orange line is cos − 1 x sin − 1 x and grey line, tan − 1 x . Note that x ∈ [ 0 , 1 ] and there are only 2 solutions, x = 0 and x ≈ 0 . 4 5 0 1 1 6 .