Maths · Inverse trigonometrical functions and their properties
Assertion(A): and are positive for all positive real values of in their domain.
Assertion(A): \( \cos ^{-1} x \) and \( \tan ^{-1} x \) are positive for all positive real values of \( x \) in their domain. Reason(R): The domain of \( f(x)= \) \( \cos ^{-1} x+\tan ^{-1} x \) is [-1,1]
- A. Both A and R are true and R is the correct explanation of A
- B. Both A and R are true but R is not correct explanation of
- C. A is true but R is false
- D. A is false but R is true
Step-by-step solution
Assertion (A): For positive x in the domain of cos⁻¹x (which is [0,1]), cos⁻¹x ranges from π/2 (at x=0) to 0 (at x=1). At x=1, cos⁻¹1 = 0, which is not positive. tan⁻¹x is positive for all positive x. Thus, the assertion that both are positive for all positive x is false. Reason (R): The domain of cos⁻¹x is [-1,1] and tan⁻¹x is ℝ, so their sum has domain [-1,1], which is true. Hence, A is false, R is true.
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