Question

sec⁴A(1-sin⁴A)-2tan²A=1 ,Prove it

Answer 1

sec^4A (1-sin^2A)(1+sin^2A) - 2tan^2A = 1
sec^2A (1+sin^2A) - 2tan^2A = 1
sec^2A (2sin^2A + cos^2A) - 2tan^2A=1
2tan^2A + 1 - 2tan^2A= 1
1=1

Answer 2

Answer:

Sec⁴A(1-sin⁴A)-2tan²A=1

Step-by-step explanation:

Sec⁴A(1-sin⁴A)-2tan²A=1

LHS = Sec⁴A(1-sin⁴A)-2tan²A

= (Sec²A)²(1-sin⁴A)-2tan²A

SecA = 1/CosA

= (1-sin⁴A)/(Cos²A)²  - 2tan²A

a² - b² = (a +b) (a -b)

= (1 + sin²A)(1-Sin²A)/(Cos²A)²  - 2tan²A

1-Sin²A = Cos²A

= (1 + sin²A)Cos²A/(Cos²A)²  - 2tan²A

using tanA = SinA/CosA

= (1 + sin²A)/Cos²A - 2Sin²A/Cos²A

=(1/Cos²A) ( 1 + sin²A - 2 Sin²A)

= (1/Cos²A)( 1 - sin²A)

= ( 1 - sin²A)/Cos²A

= Cos²A/Cos²A

= 1

=  RHS

QED

Proved