Let be an arbitrary point on the unit circle with

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Let

R(x,\,y)

be an arbitrary point on the unit circle with

x\neq0

, representing an angle

\beta

in standard position. (i) Prove that

\displaystyle \tan \beta=\dfrac{y}{x}

. (ii) Using your result from part (i) and the relation

x^{2}+y^{2}=1

, derive the identity

\displaystyle \sec^{2}\!\beta-\tan^{2}\!\beta=1

. (iii) Explain why your derivation is valid for every real angle except odd multiples of

\dfrac{\pi}{2}