Q27 / 120 · NDA 2024 (Phase 2) · Mathematics

What is the general solution of $\cos ^{100} x-\sin ^{100} x=1$ ? where $n$ is an integer.

  • Correct answer
    A
    $n \pi$
  • B
    $(2 n+1) \pi$
  • C
    $2 n \pi$
  • D
    $(2 n+1) \frac{\pi}{2}$

Solution

Since, $\cos ^2 \mathrm{x} \in[0,1] \Rightarrow \cos ^{100} \mathrm{x} \in[0,1]$ and $\sin ^2 x \in[0,1] \Rightarrow \sin ^{100} x \in[0,1]$ Hence, $\cos ^{100} \mathrm{x}-\sin ^{100} \mathrm{x}=1$ is only possible when $\begin{aligned} & \cos ^{100} x=1 \\ & \Rightarrow \cos ^2 x=1 \Rightarrow x=n \pi \pm 0 \Rightarrow x=n \pi \end{aligned}$

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

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