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Question

The maximum value of open parentheses cos space alpha subscript 1 close parentheses open parentheses cos space alpha subscript 2 close parentheses horizontal ellipsis open parentheses cos space alpha subscript n close parentheses under the restrictions 0 less or equal than alpha subscript 1 comma alpha subscript 2 horizontal ellipsis horizontal ellipsis alpha subscript n less or equal than pi over 2 and cot space alpha subscript 1 times cot space alpha subscript 2 horizontal ellipsis horizontal ellipsis cot space alpha subscript n equals 1 is

  1. 1 over 2 to the power of n divided by 2 end exponent
  2. 1 over 2 to the power of n
  3. fraction numerator 1 over denominator 2 n end fraction
  4. 1

Hint:

In the question we will first derive the possible value of the angles in order to get the value of the function for this we will use the given functions then we will simplify it to derive the value of angles.

The correct answer is: 1 over 2 to the power of n divided by 2 end exponent


    Given, cot space alpha subscript 1 times cot space alpha subscript 2 horizontal ellipsis horizontal ellipsis cot space alpha subscript n equals 1
    rightwards double arrow fraction numerator cos alpha subscript 1 over denominator sin alpha subscript 1 end fraction space. fraction numerator cos alpha subscript 2 over denominator sin alpha subscript 2 end fraction space. fraction numerator cos alpha subscript 3 over denominator sin alpha subscript 3 end fraction..... fraction numerator cos alpha subscript n over denominator sin alpha subscript n end fraction equals 1
rightwards double arrow cos alpha subscript 1. space cos alpha subscript 2. space cos alpha subscript 3...... cos alpha subscript n equals sin alpha subscript 1. space sin alpha subscript 2. space sin alpha subscript 3...... sin alpha subscript n
t h i s space i s space p o s s i b l e space o n l y space w h e n space alpha subscript 1 equals alpha subscript 2 equals space alpha subscript 3 equals..... alpha subscript n equals 45 degree
S o comma space cos alpha subscript 1. space cos alpha subscript 2. space cos alpha subscript 3...... cos alpha subscript n
equals fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 1 over denominator square root of 2 end fraction. fraction numerator 1 over denominator square root of 2 end fraction....... fraction numerator 1 over denominator square root of 2 end fraction
equals 1 over 2 to the power of begin display style n over 2 end style end exponent

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