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Question

$cos to the power of 3 end exponent invisible function application x sin invisible function application 2 x equals not stretchy sum from x equals 0 to n of a subscript r end subscript sin invisible function application open parentheses r blank x close parentheses for all x element of R comma blank t h e n$

$n = 5, a_{1} = \frac{1}{2}$
$n = 5, a_{1} = \frac{1}{4}$
$n = 5, a_{2} = \frac{1}{8}$
$n = 5, a_{2} = 1 / 4$

The correct answer is: $n equals 5 comma a subscript 1 end subscript equals fraction numerator 1 over denominator 4 end fraction$

$cos to the power of 3 end exponent invisible function application x sin invisible function application 2 x blank equals cos to the power of 2 end exponent invisible function application x cos invisible function application x sin invisible function application 2 x$
$equals open parentheses fraction numerator 1 minus cos invisible function application 2 x over denominator 2 end fraction close parentheses blank open parentheses fraction numerator 2 sin invisible function application 2 x cos invisible function application x over denominator 2 end fraction close parentheses$
$equals fraction numerator 1 over denominator 4 end fraction open parentheses 1 minus cos invisible function application 2 x close parentheses open parentheses sin invisible function application 3 x plus sin invisible function application x close parentheses$
$equals fraction numerator 1 over denominator 4 end fraction open square brackets sin invisible function application 3 x plus sin invisible function application x minus fraction numerator 1 over denominator 2 end fraction open parentheses 2 sin invisible function application 3 x. cos invisible function application 2 x close parentheses minus fraction numerator 1 over denominator 2 end fraction open parentheses 2 cos invisible function application 2 x sin invisible function application x close parentheses close square brackets$
$equals fraction numerator 1 over denominator 4 end fraction open square brackets sin invisible function application 3 x plus sin invisible function application x minus fraction numerator 1 over denominator 2 end fraction open parentheses sin invisible function application 5 x plus sin invisible function application x close parentheses minus fraction numerator 1 over denominator 2 end fraction open parentheses sin invisible function application 3 x minus sin invisible function application x close parentheses close square brackets$
$equals fraction numerator 1 over denominator 4 end fraction open square brackets sin invisible function application x plus fraction numerator 1 over denominator 2 end fraction sin invisible function application 3 x minus fraction numerator 1 over denominator 2 end fraction sin invisible function application 5 x close square brackets$
$rightwards double arrow a subscript 1 end subscript equals 1 divided by 4 comma blank a subscript 3 end subscript equals 1 divided by 8 comma blank n equals 5$

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