Answer with explanation:
We have to find that function,which has range ,(¬∞,0)∪(2,∞) out of the following four trigonometric functions.
Range of a function is defined as,those values of function,if the function is →y=f(x), that is those values of y , for which x is defined.
A:→ y =sec x
x=[tex]sec^{-1}y[/tex]
x is defined for , y≥1 and y≤-1.
B: →y=cot (2 x)-1
y+1=cot (2 x)
[tex]x=\frac{1}{2}*cot^{-1}(y+1)[/tex]
Since cotangent function is defined for all values of y, so, x∈[-∞,∞],that is ,x∈R.
So, Cot(2 x) -1, have the same range as Cot x.
C: y=Cos (x+1)
[tex]x+1=Cos^{-1}y\\\\x=Cos^{-1}y-1[/tex]
[tex]-1\leq Cos^{-1}y\leq 1\\\\-1-1\leq Cos^{-1}y-1\leq 1-1\\\\-2\leq Cos^{-1}y-1\leq 0[/tex]
D: y=Cosec (x)+1
Range of Cosec x is, (-∞,-1] ∪ [1,∞).
Range of Cosec (x) +1 will be , (-∞,-1+1] ∪ [1+1,∞).
= (-∞, 0] ∪ [2,∞)
None of the option is true.
