Number of solutions of the equation 2 cot -1 2+ cos -1(3 / 5)= operatornamecosec-1 x is

Question

Number of solutions of the equation 2 cot -1 2+ cos -1(3 / 5)= operatornamecosec-1 x is (A) 0 (B) 1 (C) 2 (D) more than 2

Tardigrade Admin · Accepted Answer

tan -1(1 / 2)+ tan -1(4 / 3)= tan -1 (2 ⋅ (1/2)/1-(1)4)+ tan -1 (4/3)=2 tan -1 (4/3)>(π/2) text but operatornamecosec-1 x ∈[-(π/2), 0) ∪(0, (π/2)] ⇒ text no solution

Q. Number of solutions of the equation $2 cot^{- 1} 2 + cos^{- 1} (3/5) = cosec^{- 1} x$ is

0

1

2

more than 2

$tan^{- 1} (1/2) + tan^{- 1} (4/3) = tan^{- 1} \frac{2 \cdot \frac{1}{2}}{1 - \frac{1}{4}} + tan^{- 1} \frac{4}{3} = 2 tan^{- 1} \frac{4}{3} > \frac{π}{2}$
$but cosec^{- 1} x \in [- \frac{π}{2}, 0) \cup (0, \frac{π}{2}] \Rightarrow no solution$

Q. Number of solutions of the equation 2cot−12+cos−1(3/5)=cosec−1x is

0

1

2

more than 2

tan−1(1/2)+tan−1(4/3)=tan−11−41​2⋅21​​+tan−134​=2tan−134​>2π​ but cosec−1x∈[−2π​,0)∪(0,2π​]⇒ no solution

Q. Number of solutions of the equation $2 cot^{- 1} 2 + cos^{- 1} (3/5) = cosec^{- 1} x$ is

$tan^{- 1} (1/2) + tan^{- 1} (4/3) = tan^{- 1} \frac{2 \cdot \frac{1}{2}}{1 - \frac{1}{4}} + tan^{- 1} \frac{4}{3} = 2 tan^{- 1} \frac{4}{3} > \frac{π}{2}$
$but cosec^{- 1} x \in [- \frac{π}{2}, 0) \cup (0, \frac{π}{2}] \Rightarrow no solution$