If $$cos^{4⁡}\theta-sin^{4⁡}\theta=\frac 45$$​, then find the value of sin 4θ.

Correct option is A

Given:
$$\cos^4 \theta - \sin^4 \theta = \frac{4}{5}\\$$​
Formula Used:
​$$(a^2 - b^2) = (a + b)(a - b)\\\cos^2 \theta + \sin^2 \theta = 1\\\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 1 - 2\sin^2 \theta\\\sin 2\theta = 2\sin\theta\cos\theta\\$$​​
Solution:

$$\cos^4 \theta - \sin^4 \theta = \frac{4}{5}\\(\cos^2 \theta - \sin^2 \theta)(\cos^2 \theta + \sin^2 \theta) = \frac{4}{5}\\\cos^2 \theta - \sin^2 \theta = \frac{4}{5} \quad [\because \cos^2 \theta + \sin^2 \theta = 1]\\1 - \sin^2 \theta - \sin^2 \theta = \frac{4}{5} \quad [\because \cos^2 \theta = 1 - \sin^2 \theta]\\1 - 2\sin^2 \theta = \frac{4}{5}\\\cos 2\theta = \frac{4}{5}\\\sin 2\theta = \frac{3}{5} \quad [h=5, \, b=4, \, p=3 \, \text{using Pythagoras' theorem}]\\\sin 4\theta = 2 \sin 2\theta \cos 2\theta\\= 2 \times \frac{3}{5} \times \frac{4}{5}\\= \frac{24}{25}\\$$​