Question

$\cos^{4} \theta - \sin {}^{4} \theta = \frac{2}{3} \: \: then \: \cos {}^{2} \theta - 1 =$ ?​

Answer 1

Question:

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\cos^{4} \theta - \sin {}^{4} \theta = \frac{2}{3} \: \: then \: \cos {}^{2} \theta - 1 =$ ?

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Solution:

⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

We know that,

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\sin {}^{2} \theta + \cos {}^{2} \theta = 1$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\cos {}^{4} \theta - \sin {}^{4} \theta = \frac{2}{3}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$( \cos {}^{2}\theta + \sin {}^{2} \theta )( \cos {}^{2} \theta - \sin {}^{2} \theta ) = \frac{2}{3}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow \sin {}^{2} \theta = 1 - \cos {}^{2} \theta$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow \cos {}^{2} \theta - (1 - \cos {}^{2} \theta) = \frac{2}{3}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\longrightarrow2 \cos {}^{2} \theta - 1 = \frac{2}{3}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Thus, 2/3 is the required solution.