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Answers
Since sin x and cosec x are reciprocals to each other,
let sin x = a and cosec x = 1/a.
And, since the RHS is 2, we can use the concept given below.
If and only if a + 1/a = 2, aⁿ + 1/aⁿ will also be equal to 2, for any real number n.
Why this concept always holds true?!
a + 1/a = 2
=> a(a + 1/a) = 2a
=> a² + 1 = 2a
=> a² - 2a + 1 = 0
=> (a - 1)² = 0
=> a = 1
So,
aⁿ + 1/aⁿ = 1ⁿ + 1/1ⁿ = 1 + 1 = 2
Hence that's why.
Thus, we can say,
sinⁿ x + cosecⁿ x= 2
Taking n = 19.5,
sin^19.5 x + cosec^19.5 x = 2
=> cosec^0.5 x(sin^19.5 x + cosec^19.5 x) = 2 cosec^0.5 x
=> ((sin^19.5 x)/(sin^0.5 x)) + cosec^(19.5 + 0.5) x = 2 cosec^0.5 x
=> sin^19 x + cosec^20 x = 2√(cosec x)
Okay, we got 2√(cosec x) as answer.
First we took sin x and cosec x as a and 1/a respectively, them we got a = 1, thereby getting 1/a = 1.
Thus we get cosec x = 1.
=> 2√(cosec x) = 2
Or we can directly take both sin x and cosec x as 1, this avoids the above large solution.
Since sin x = cosec x = 1,
sin^19 x + cosec^20 x = 1^19 + 1^20 = 1 + 1 = 2
Hence, 2 is the answer.
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