The expression (tan⁶θ-sec⁶θ+3sec²θ tan²θ)/(tan² θ + cot² θ +2), 0⁰, is equal to:

The expression (tan⁶θ-sec⁶θ+3sec²θ tan²θ)/(tan² θ + cot² θ…

The expression $\frac{tan^6\theta-sec^6\theta+3sec^2\theta tan^2\theta}{tan^2 \theta + cot^2 \theta +2}, 0^0<\theta<90^0$ , is equal to:

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Correct answer: D

Explanation

The correct answer is option 4 i.e. -cos² $\theta$ sin² $\theta$

Shortcut method :

Put $\theta$ = 45°.(because 0° < $\theta$ < 90°)

Sec45 = $\sqrt2$ , tan45= 1, cot45 = 1, cot45 = 1and cosec 45 = $\sqrt2$

= (1 - 8 + 3 × 2 × 1)/( 1 + 1 + 2)

= -1/4.

Now put $\theta$ = 45 in the option as well.

Option 1 = 2 × 2 = 4 not matches with our answer.

Option 2 = -2 × 2 = -4 (not matches with our answer)

Option 3 = 1/2 × 1/2 = 1/4 ( not matches with our answer)

Option 4 = -1/2 × 1/2 = -1/4 . matches with the answer.