Q: Indicate the relation which is false

(A) tan | tan^{-1} x | = | x |

(B) cot | cot^{-1} x | = x

(C) tan^{-1} | tan x | = | x |

(D) sin | sin^{-1} x | = | x |

Sol. Since $ \large | tan^{-1}x | = \left\{\begin{array}{ll} tan^{-1}x \; , if \; 0 \leq tan^{-1}x \le \pi/2 \\ -tan^{-1}x \; , if \; -\pi/2 < tan^{-1}x < 0 \end{array} \right. $

$ \large = \left\{\begin{array}{ll} tan^{-1}x \; , if \; x \ge 0 \\ -tan^{-1}x \; , if x < 0 \end{array} \right. $

⇒ | tan^{-1}x | = tan^{-1} |x| ∀ x ∈ R

⇒ tan| tan^{-1}x | = tan tan^{-1} |x| = |x|

Hence (A) is the correct answer.

Likewise sin | sin^{-1}x| = sin sin^{-1}|x| = |x| ∀ |x| ≤ 1

Hence (D) is the correct answer.

| cot^{-1}x | = cot^{-1}x as 0 < | cot^{-1}x| < π ∀ x ∈ R

⇒ cot |cot^{-1}x| = cot cot^{-1}x = x .

Hence (B) is the correct answer.

Since | tanx | is not necessarily always equal to tan|x|

Hence tan^{-1} | tanx | ≠ | x| .