Question 12 Marks
Prove the following.
Show that $\tan^4\theta+\tan^2\theta=\sec^4\theta-\sec^2\theta.$
Show that $\tan^4\theta+\tan^2\theta=\sec^4\theta-\sec^2\theta.$
Answer
View full question & answer→LHS $\tan^4\theta+\tan^2\theta=\tan^2\theta(\tan^2\theta+1)$
$=\tan^2\theta\cdot\sec^2\theta\ [\therefore\sec^2\theta=\tan^2\theta+1]$
$=(\sec^2\theta-1)\cdot\sec^2\theta\ [\therefore\tan^2\theta=\sec^2\theta-1]$
$=\sec^4\theta-\sec^2\theta=$ RHS
$=\tan^2\theta\cdot\sec^2\theta\ [\therefore\sec^2\theta=\tan^2\theta+1]$
$=(\sec^2\theta-1)\cdot\sec^2\theta\ [\therefore\tan^2\theta=\sec^2\theta-1]$
$=\sec^4\theta-\sec^2\theta=$ RHS