Q171 / 270 · NDA 2020 · Mathematics

Paragraph: Read the following information and answer Let $t_n=\sin ^n \theta+\cos ^n \theta$.
Question: What is the value of $t_{10}$ where $\theta=45^{\circ}$ ?

  • A
    1
  • B
    $\frac{1}{4}$
  • Correct answer
    C
    $\frac{1}{16}$
  • D
    $\frac{1}{32}$

Solution

$t_{10}=\sin ^{10}(\theta)+\cos ^{10}(\theta)$ When $\theta=45^{\circ}$, $t_{10}=\sin ^{10}\left(45^{\circ}\right)+\cos ^{10}\left(45^{\circ}\right)$ $\begin{aligned} & =\left(\frac{1}{\sqrt{2}}\right)^{10}+\left(\frac{1}{\sqrt{2}}\right)^{10}=2 \cdot\left(\frac{1}{\sqrt{2}}\right)^{10} \\ & =2 \times \frac{1}{32}=\frac{1}{16}\end{aligned}$

Hence, the correct answer is option: 'C'.

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