Question

if A = cos²A+ sin⁴∅, then prove that fot all values of ∅, 3/4<= A <= 1

Answer 1

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Answer 2

Also Cos^2x >= Cos^4x

Add sin^2x both sides

Cos^2x + sin^2x >= Cos^4x + sin^2x

1 >= Cos^4x + sin^2x

Therefore max of A is 1

Also both terms  

 

Cos^4x + sin^2x

= (1 – sin^2x)^2 + sin^2x

= 1 + sin^4x – 2sin^2x + sin^2x

=Sin^4x – sinn^2x + 1

= (sin^2x – 1/2)^2  + 3/4

now Min =3/4 and Max = 1