- Home
- Standard 11
- Mathematics
Trigonometrical Equations
normal
If $cosx + secx =\, -2$, then for a $+ve$ integer $n$, $cos^n x + sec^n x$ is
A
always $2$
B
always $-2$
C
$-2$ if $n$ is odd and $2$ if $n$ is even
D
$-2$ if $n$ is even and $2$ if $n$ is odd
Solution
$\cos x+\sec x=-2$
$\Rightarrow \cos x+\frac{1}{\cos x}=-2$
$\Rightarrow \frac{\cos ^{2} x+1}{\cos x}=-2$
$\Rightarrow \cos ^{2} x+1=-2 \cos x$
$\Rightarrow \cos ^{2} x+2 \cos x+1=0$
$\Rightarrow(\cos x+1)^{2}=0 \quad \Rightarrow \quad \cos x=-1$
$\sin x =\sqrt{1-\cos ^{2} x}$
$=\sqrt{1-1}=0$
$\cos x=-1, \sin x=0$
$\cos ^{n} x+\sin ^{n} x=(-1)^{n}+0$
$\left\{\begin{array}{cc}-1 & n \text { is odd } \\ 1 & n \text { is even }\end{array}\right.$
Standard 11
Mathematics
