Question

Show that Cot⁴ A+ Sin⁴+2Sin ²A Cos² A=1​

Answer 1

Answer:

Replacing theta with A

1+cos²2A=1+(2cos²A-1)²=1+4cos⁴A-4cos²A+1=2+4(cos⁴A-cos²A)=2(sin²A+cos²A) + 4cos⁴A-4cos²A

=2sin²A - 2cos²A + 4cos⁴A

=2sin²A - 2cos²A + 4(cos ²A)²

=2sin²A - 2cos²A + 4(1 - sin²A)²

=2sin²A -2cos²A + 4(1 - 2sin²A + sin⁴A)

=2sin²A - 2cos²A + 4 - 8sin²A + 4sin⁴A

=2sin²A - 2cos²A + 4 cos²A + 4sin²A - 8sin²A+ 4 sin⁴A

=2cos²A - 2sin²A + 4sin⁴A

=2cos2A + 4 sin⁴A

=2(1- 2sin²A) + 4sin⁴A

= 2 - 4sin²A + 4sin⁴A

=2 - 4sin²A + 2sin⁴A + 2sin⁴A

=2( 1- 2sin²A + sin⁴A) + 2sin⁴A

=2( 1 - sin²A)² + 2 sin⁴A

=2cos⁴A + 2 sin⁴A

=2(cos⁴A + sin⁴A)