Question

If ${\sin ^{ - 1}}(x - \dfrac{{{x^2}}}{2} + \dfrac{{{x^3}}}{4} - .....\infty ) + {\cos ^{ - 1}}({x^2} - \dfrac{{{x^4}}}{2} + \dfrac{{{x^6}}}{4}....\infty ) = \dfrac{\pi }{2}$ for $0 < \left| x \right| < \sqrt 2$ , then x equals
A.$\dfrac{1}{2}$
B.$1$
C.$\dfrac{{ - 1}}{2}$
D.$- 1$

Answer 1

Hint : First, identify the geometric progressions in the question and then find their sum using a suitable formula. Using the inverse trigonometric identities find out the relation between the functions of sin and cos. Keep in mind the conditions for x to axis. This way you can find the correct answer.

Complete step-by-step answer :
$x - \dfrac{{{x^2}}}{2} + \dfrac{{{x^3}}}{4}.....$ is a Geometric progression.
Its common ratio, $r = \dfrac{{\dfrac{{ - {x^2}}}{2}}}{x} = \dfrac{{ - x}}{2}$
The sum of an infinite geometric progression is given by the formula, $S = \dfrac{{{a_1}}}{{1 - r}}$ , where ${a_1}$ is the first term of G.P.
So the sum of given G.P. is
$\Rightarrow {S_1} = \dfrac{x}{{1 - ( - \dfrac{x}{2})}} = \dfrac{x}{{1 + \dfrac{x}{2}}} = \dfrac{{2x}}{{2 + x}}$
${x^2} - \dfrac{{{x^4}}}{2} + \dfrac{{{x^6}}}{4}.....$ is also a G.P.
Its common ratio is , $r = \dfrac{{ - \dfrac{{{x^4}}}{2}}}{{{x^2}}} = - \dfrac{{{x^2}}}{2}$
Sum of this infinite G.P. is
$\Rightarrow {S_2} = \dfrac{{{x^2}}}{{1 - ( - \dfrac{{{x^2}}}{2})}} = \dfrac{{{x^2}}}{{1 + \dfrac{{{x^2}}}{2}}} = \dfrac{{2{x^2}}}{{2 + {x^2}}}$
Now, the equation given in the question can be rewritten as,
${\sin ^{ - 1}}(\dfrac{{2x}}{{2 + x}}) + {\cos ^{ - 1}}(\dfrac{{2{x^2}}}{{2 + {x^2}}}) = \dfrac{\pi }{2}$
Now, we know that ${\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \dfrac{\pi }{2}$ when $- 1 \leqslant x \leqslant 1$
So using this identity, we get
$\Rightarrow \dfrac{{2x}}{{2 + x}} = \dfrac{{2{x^2}}}{{2 + {x^2}}} \\\ \Rightarrow 2x(2 + {x^2}) = 2{x^2}(2 + x) \\\ \Rightarrow 4x + 2{x^3} = 4{x^2} + 2{x^3} \\\ \Rightarrow 4{x^2} - 4x = 0 \\\ \Rightarrow 4x(x - 1) = 0 \;$
This gives $x = 0$ or $x = 1$
But we are given the question that $x > 0$ .
Therefore, $x = 0$ is rejected and $x = 1$ is the right answer.
So, the correct answer is “Option B”.

Note : A geometric progression is a sequence of numbers in which each term is found by the multiplication of its previous term with a non-one number, this non-one number is called the common ratio. For example, the sequence 2, 4, 8, 16,32…. Is a geometric progression or geometric sequence and the common ratio can be obtained by dividing any two consecutive numbers, in the given example the common ratio is 2.