Question

True or False ? explain your answer

18.csc^{-1} (csc(-\frac{\pi}{4} )) = -\frac{\pi}{4}
19.sec(sec^{-1} (\sqrt{3} ))=\sqrt{3}

Answer 1

`\bf 18)\\\\ csc\left(-(\pi )/(4) \right)=\stackrel{some~value}{y}\qquad therefore\qquad csc^(-1)(y)=-(\pi )/(4) \\\\\\ 19)\\\\ sec^(-1)(√(3))=\stackrel{some~angle}{\theta }\qquad therefore\qquad sec(\theta )=√(3)`

you can think of such combinations this way like in this example,

sin(90°) = 1, well, then what is sin⁻¹(1)?

well, sin⁻¹(1) is really asking "what is the angle or angles whose sine is 1?"

well, we already know that sin(90°) is 1 thus sin⁻¹( sin(90°) ) = 90°.