If cosx + secx = , -2, then for a +ve integer n, cos^n x + sec^n … - Vidyadip

MCQ

If $cosx + secx =\, -2$, then for a $+ve$ integer $n$, $cos^n x + sec^n x$ is

A
always $2$
B
always $-2$
✓
$-2$ if $n$ is odd and $2$ if $n$ is even
D
$-2$ if $n$ is even and $2$ if $n$ is odd

✓

Answer

Correct option: C.

$-2$ if $n$ is odd and $2$ if $n$ is even

c
$\cos x+\sec x=-2$

$\Rightarrow \cos x+\frac{1}{\cos x}=-2$

$\Rightarrow \frac{\cos ^{2} x+1}{\cos x}=-2$

$\Rightarrow \cos ^{2} x+1=-2 \cos x$

$\Rightarrow \cos ^{2} x+2 \cos x+1=0$

$\Rightarrow(\cos x+1)^{2}=0 \quad \Rightarrow \quad \cos x=-1$

$\sin x =\sqrt{1-\cos ^{2} x}$

$=\sqrt{1-1}=0$

$\cos x=-1, \sin x=0$

$\cos ^{n} x+\sin ^{n} x=(-1)^{n}+0$

$\left\{\begin{array}{cc}-1 & n \text { is odd } \\ 1 & n \text { is even }\end{array}\right.$

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