Question 62

The expression $$3 \sec^2 \theta \tan^2 \theta + \tan^6 \theta - \sec^6 \theta$$ is equal to:

Solution

$$3 \sec^2 \theta \tan^2 \theta + \tan^6 \theta - \sec^6 \theta$$

= $$-3 \sec^2 \theta \tan^2 \theta(\tan^2 \theta - \sec^2 \theta) + \tan^6 \theta - \sec^6 \theta$$

($$\because \tan^2 \theta - \sec^2 \theta = -1$$)

= ($$\tan^2 \theta - \sec^2 \theta$$)^3

($$\because(a - b)^3 = a^3 - b^3 - 3ab(a - b)$$)

= $$(-1)^3$$ = -1

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