Mathematics

Problem

Let

x\in[-1,1]

. Consider the trigonometric expressions that involve both direct and inverse functions. a. (i) Prove that

\cos\bigl(\arcsin x\bigr)=\sqrt{1-x^{2}}

for all

x

in the given domain. b. (ii) Using the result of part (i), show that

\displaystyle \tan\bigl(\arcsin x\bigr)=\dfrac{x}{\sqrt{1-x^{2}}}

. c. (iii) Hence evaluate (a)

\cos\bigl(\arcsin\tfrac{3}{5}\bigr)

and (b)

\tan\bigl(\arcsin(-\tfrac45)\bigr)

. Express each answer as a rational number or a simplified radical.