Question

Prove that $si{n}^4\theta -co{s}^4\theta =1-2co{s}^2\theta$ [3 MARKS]

Answer 1

Formula: 2 Marks
Application: 1 Mark

Consider

$si{n}^4\theta -co{s}^4\theta =(si{n}^2\theta {)}^2-(co{s}^2\theta {)}^2$

$a^2-b^2=(a+b)(a-b)$

$=(si{n}^2\theta +co{s}^2\theta )(si{n}^2\theta -co{s}^2\theta )$

$=1(si{n}^2\theta -co{s}^2\theta )$ $[\because si{n}^2\theta +co{s}^2\theta =1]$

$=si{n}^2\theta -co{s}^2\theta$

$=(1-co{s}^2\theta )-co{s}^2\theta$ $[si{n}^2\theta =1-co{s}^2\theta ]$

$=1-2co{s}^2\theta$