Question

Which of the following trigonometric expressions is equal to ${\text{sec}}^6\theta$ ?

• A. ${\text{tan}}^6\theta +3{\text{tan}}^2\theta {\text{sec}}^2\theta +1$
• B. ${\text{tan}}^6\theta -3{\text{tan}}^2\theta {\text{sec}}^2\theta +1$
• C. ${\text{tan}}^6\theta -1$
• D. ${\text{tan}}^6\theta +1$

Answer 1

The correct option is A

${\text{tan}}^6\theta +3{\text{tan}}^2\theta {\text{sec}}^2\theta +1$

${\text{sec}}^6\theta =({\text{sec}}^2\theta {)}^3$

$=({\text{tan}}^2\theta +1{)}^3\dots \dots (1+{\text{tan}}^2\theta ={\text{sec}}^2\theta )$

$={\text{tan}}^6\theta +3{\text{tan}}^4\theta +3{\text{tan}}^2\theta +1$

$={\text{tan}}^6\theta +3{\text{tan}}^2\theta \left({\text{tan}}^2\theta \right)+3{\text{tan}}^2\theta +1$

$={\text{tan}}^6\theta +3{\text{tan}}^2\theta ({\text{sec}}^2\theta -1)+3{\text{tan}}^2\theta +1$

$={\text{tan}}^6\theta +3{\text{tan}}^2\theta {\text{sec}}^2\theta -3{\text{tan}}^2\theta +3{\text{tan}}^2\theta +1$

$={\text{tan}}^6\theta +3{\text{tan}}^2\theta {\text{sec}}^2\theta +1$