Mathematics

Problem

Let

\theta

be any real angle. Use the definitions of the trigonometric ratios to explore a fundamental relationship. a. (i) Express

\tan\theta

and

\sec\theta

entirely in terms of

\sin\theta

and

\cos\theta

. No other trigonometric ratio should appear in your expressions. b. (ii) Starting from the identity

\sin^{2}\theta+\cos^{2}\theta=1

and the results of part (i), prove that

1+\tan^{2}\theta=\sec^{2}\theta

for all values of

\theta

for which both sides are defined. c. (iii) If

\cos\theta=\dfrac{5}{13}

and

\theta

lies in the fourth quadrant, evaluate

1+\tan^{2}\theta

numerically and verify that it equals

\sec^{2}\theta

for this value of

\theta

. Show every step of your calculation.