Which of following pairs of functions are identical.
and g(x) = sec − 1 x (b ) f(x) = tan (tan − 1 x) and g(x) = cot (cot − 1 x) (c ) f(x) = sgn (x) and g(x) = sgn (sgn (x)) (d ) f(x) = cot 2 x. cos 2 x and g(x) = cot 2 x − cos 2 x
Text Solution
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(b,c,d)
f(x) = 
= sec –1 x, x ∈ (– ∞ , – 1] ∪ (1, ∞ )
g(x) = sec –1 x, x ∈ (– ∞ , – 1] ∪ [1, ∞ )
non-identical functions
f(x) = tan (tan –1 x) = x, x ∈ R
g(x) = cot (cot –1 x) = x, x ∈ R
identical functions
f(x) = sgn (x) = 
⇒ g(x) = sgn(sgn x) = 
Identical functions
f(x) = cot 2 x . cos 2 x, x ∈ R – {n π }, n ∈ I
g(x) = cot 2 x – cos 2 x
= cot 2 x (1 – sin 2 x) = cot 2 x. cos 2 x x ∈ R – {n π }, n ∈ I
Identical functions
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