Question

plz answer to this question as soon as possible plz.......................​

Answer 1

Question:-

$\cos ^ { 4 } \alpha + 2 \cos ^ { 2 } \alpha ( 1 - \frac { 1 } { \sec ^ { 2 } \alpha } ) = ( 1 - \sin ^ { 4 } \alpha )$

Answer:-

$LHS = \: \cos ^ { 4 } \alpha + 2 \cos ^ { 2 } \alpha ( 1 - \frac { 1 } { \sec ^ { 2 } \alpha } ) = ( 1 - \sin ^ { 4 } \alpha ) \\ \\ = cos {}^{4} \alpha + 2cos {}^{2} \alpha (1 - cos {}^{2} \alpha ) \\ \\ = cos {}^{4} \alpha + 2cos {}^{2} \alpha - 2cos {}^{4} \alpha \\ \\ = \: \: 2cos {}^{2} \alpha - cos {}^{4} \alpha \\ \\ = cos {}^{2} \alpha (2 - cos {}^{2} \alpha ) \\ \\ = (1 - sin {}^{2} \alpha )(2 - 1 + sin {}^{2} ) \\ \\ = (1 - sin {}^{2} \alpha )(1 + sin {}^{2} \alpha ) \\ \\ = 1 - sin {}^{4} \alpha = RHS$