Question

how to prove that sin^8@ + cos^8@
= (1-2cos^2@)(1 - 2sin^2@ cos^2@)

Answer 1

Answer:

hey mate here is ur answer

Step-by-step explanation:

Hey there !

starting with LHS :

=> sin⁸A - cos⁸A

we know that :-

sin⁸A = (sin⁴A)² 

cos⁸A = (cos⁴A)²

This can be written as :-

 => (sin⁴A)² - (cos⁴A)²

Now this is in the form of an identity : a² - b² = (a+b) ( a - b)

 => (sin⁴A + cos⁴A) ( sin⁴A - cos⁴A)

sin⁴A  = (sin²A)²

cos⁴A = (cos²A)²

=>  (sin²A)² + (cos²A)² (( sin⁴A - cos⁴A))

 (sin²A)² + (cos²A)² can be in the identity : a² + b² = (a+b)² - 2ab

  [    (sin²A)² + (cos²A)² =  (sin²A + cos²A)² - 2sin²A cos²A ]

 => [(sin²A + cos²A)² - 2sin²A cos²A ] (( sin⁴A - cos⁴A))

Now ,

sin⁴A - cos⁴A 

this can be written in the form of the identity a² - b² = (a+b) (a -b)

sin⁴A  = (sin²A)²

cos⁴A = (cos²A)²

 sin⁴A - cos⁴A = (sin²A + cos²A) (sin²A - cos²A)

=> => [(sin²A + cos²A)² - 2sin²A cos²A ] (sin²A + cos²A) (sin²A - cos²A)

we know that ,

sin²A + cos²A = 1              [ by identity ]

hence,

=> [ (1)² - 2sin²A cos²A ] (1) ×(sin²A - cos²A)

=> ( 1 - 2sin²A cos²A ) (sin²A - cos²A)

=>RHS

hope this will help u plzzzzz mark me as brainlist