Question

If $sin\theta +cos\theta =a$ and $sec\theta +cosec\theta =b$, then what is the value of $b(a^2-1)$? [3 MARKS]

Answer 1

Concept: 1 Mark
Application: 2 Marks

$b(a^2-1)$

$=(sec\theta +cosec\theta )\left(\right(sin\theta +cos\theta {)}^2-1)$

$=(sec\theta +cosec\theta )(si{n}^2\theta +co{s}^2\theta +2sin\theta cos\theta -1)$

$=(sec\theta +cosec\theta )(1+2sin\theta cos\theta -1)$

$=(sec\theta +cosec\theta )\left(2sin\theta cos\theta \right)$

$=2sin\theta cos\theta sec\theta +2sin\theta cos\theta cosec\theta$

$=2sin\theta +2cos\theta$

$=2(sin\theta +cos\theta )$

$=2a$