Mathematics

Problem

Prove that the following expression is a fixed number for every acute angle

\theta

F(\theta)=\tan(90^{\circ}-\theta)\,\tan\theta+\cot(90^{\circ}-\theta)\,\cot\theta

Present your work in the sequence given below. a. (a) Show that

\tan(90^{\circ}-\theta)\,\tan\theta=1

. b. (b) Show that

\cot(90^{\circ}-\theta)\,\cot\theta=1

. c. (c) Hence find the numerical value of

F(\theta)

. d. (d) Determine

[F(\theta)]^{2}

and comment on whether it depends on

\theta