Question

sin4 a+2sin²a[1-1/cosec²a]
-cos4 a = ......
​

Answer 1

Answer:

Step-by-step explanation:

Sin⁴A + 2Sin²A ( 1 - 1/Cosec²A) - Cos⁴A

=> Sin⁴A + 2Sin²A (1 - Sin²A) - Cos⁴A

=> Sin⁴A - Cos⁴A + 2Sin²ACos²A (∵ 1 - Sin²A = Cos²A)

=> (Sin²A)² - (Cos²A)² + 2Sin²ACos²A

//  (Sin²A)² - (Cos²A)² is of form a² - b²; But a² - b² = (a + b) (a - b)

=> (Sin²A + Cos²A) (Sin²A - Cos²A) + 2Sin²ACos²A (∵ Sin²A + Cos²A = 1)

=> Sin²A - Cos²A + 2Sin²ACos²A  

//from here you can have multiple ways to give the solution:

Option 1 :                                              Option2:

=>1 - 2Cos²A + 2Sin²A Cos²A  |   => 1/2[4SinACosA]² - (Cos²A - Sin²A)  

=> 1 - 2Cos²A ( 1 - Sin²A)          |   => 1/2 * Sin²2A - Cos2A

=> 1 - 2Cos⁴A                            |   => SIn²2A/2 - Cos2A.