Mathematics

Problem

Answer the following questions about fundamental trigonometric identities. a. (i) Starting with

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

, derive the relation

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

. b. (ii) Using a similar method, derive

\csc^{2}\theta-\cot^{2}\theta=1

from the primary identity. c. (iii) Explain, in words and symbols, why the expressions

\sec^{2}\theta-\tan^{2}\theta

and

\csc^{2}\theta-\cot^{2}\theta

are each equal to a constant for every permissible value of

\theta