Question

Show that sin^4 A -cos^4A = 1 - 2 cos^2A PLZZ sove this​

Answer 1

Step-by-step explanation:

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Answer 2

$\bold{\pink{To \: prove }}\longrightarrow\\ {sin}^{4} \alpha - {cos}^{4} \alpha = 1 - 2 {cos}^{2} \alpha$

$\bold{\green{Concept \: used}}\longrightarrow \\ 1) {a}^{2} - {b}^{2} = (a + b) \: (a - b) \\ 2) {sin}^{2} \alpha + {cos}^{2} \alpha = 1 \\ 3) {sin}^{2} \alpha = 1 - {cos}^{2} \alpha$

$\bold{\blue{Solution}}\longrightarrow \:\pink{ LHS }\\ = {sin}^{4} \alpha - {cos}^{4} \alpha \\ = { ({sin}^{2} \alpha) }^{2} - {( {cos}^{2} \alpha )}^{2} \\ applying \: ( {a}^{2} - {b}^{2} ) = (a + b) \: (a - b) \\ = ( {sin}^{2} \alpha + {cos}^{2} \alpha ) \: ( {sin}^{2} \alpha - {cos}^{2} \alpha ) \\ now \: applying \: second \: and \: third \: formulee \\ = ( \: 1 \: ) \: (1 - {cos}^{2} \alpha - {cos}^{2} \alpha ) \\ = 1 - 2 {cos}^{2} \alpha \\ =\pink{ RHS}$

$\bold{\red{Additional \: information}}\longrightarrow \\ 1)1 + {tan}^{2} \alpha = {sec}^{2} \alpha \\ 2)1 + {cot}^{2} \alpha = {cosec}^{2} \alpha$