Answer the following questions about fundamental trigonometric identities.

MATHEMATICS · Multi-part · CBSE · Mathematics · Introduction to Trigonometry · Introduction to Trigonometry

Problem

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

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

, derive the relation

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

. (ii) Using a similar method, derive

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

from the primary identity. (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