Question

Cos^4 theta-sin^4 theta =2/3 then the value of 2 cos2 theta-1 is

Answer 1

Hope u get it .......................

Answer 2

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

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

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

$\underline {\blue { By \: Algebraic I\:dentity : }}$

$\boxed { \pink { a^{2} - b^{2} = ( a - b )( a + b ) }}$

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

$\underline {\blue { By \: Trigonometric\:dentity : }}$

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

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

$\implies cos^{2} \theta - 1 + cos ^{2} \theta) = \frac{2}{3}$

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

Therefore.,

$\red { Value \:of\: 2cos^{2} \theta - 1 }\green {= \frac{2}{3} }$

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