Question

How many false relations out of the following relations?

(1) tan | tan⁻¹ x| = |x|,

(2) cot| cot⁻¹ x| = |x|,

(3) tan⁻¹ |tan x| = |x|,

(4) sin | sin−1x| = |x

With the help of:

Answer 1

(1) Domain of the function tan |tan⁻¹ x| is \(\big(-\cfrac{\pi}2,\cfrac{\pi}2\big).\)

(\(\because\) tan(-θ) = - tan θ, tan tan^-1x = x and tan^-1 x ≥ 0 only if x ≥ 0 in the domain (\(-\cfrac{\pi}2,\cfrac{\pi}2\)) )

= | x |.

Therefore this relation is true.

(2) Domain of the function cot |cot^-1x| = |x| is (0, π).

(\(\because\) cot(-θ) = cot θ and cot^-1x ≥ 0 only if x ≤ \(\cfrac{\pi}2\) in the domain(0, π))

= x where x ∈  (0, π ). ( \(\because\) cot cot⁻¹ x = x )

And we know that |x| = x, when x ∊ (0, π).

Therefore this relation is true.

(3) Domain of the function tan⁻¹ |tan x| = |x| is R − {(2n + 1)\(\cfrac{\pi}2\)}.

Since, |x| is a function and we know that tan⁻¹ tan x forms a function if we restrict the range of function tan⁻¹ x in (− \(\cfrac{\pi}2\) , \(\cfrac{\pi}2\) ) otherwise tan⁻¹ tan 0 gives images 0 and nπ  where n ∈ Z which is not function according to the definition.

Therefore, we restrict the range of tan^-1x in \((-\cfrac{\pi}2,\cfrac{\pi}2)\).

Now, take x = π

⇒ tan x = tan π = 0

⇒ tan⁻¹ tan = tan⁻¹ tan π = tan⁻¹ 0 = 0 ≠ π.

Therefore, tan⁻¹ tan x ≠ x

Therefore, tan⁻¹ tan x ≠ |x|.

Hence, this relation is false.

(4) Domain of the function sin | sin⁻¹ x| = |x| is \(\big[-\cfrac{\pi}2,\cfrac{\pi}2\big].\)

(\(\therefore\) cot(-θ) = cot θ and sin^-1x ≥ 0 only if x ≥ 0 in the domain \(\big[-\cfrac{\pi}2,\cfrac{\pi}2\big]\))

Therefore this relation is true.

Hence, total number of false relation is 1.