The set of all values of \[\lambda\] for which the equation \[{\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda \] has a real solution x, is :
Solutions · 1
Official solution
The given equation is
\[\cos ^2 2x - 2 \sin ^4 x - 2 \cos ^2 x = \lambda\]
Using the trigonometric identities \[\cos^2x = 1 - \sin^2x\] and \[\cos^22x = 1 - 2\sin^2x\], we can rewrite the equation in terms of \[\cos^2x\]:
\[\lambda = (2 \cos ^2 x - 1)^2 - 2(1 - \cos ^2 x)^2 - 2 \cos ^2 x\]
Simplify this equation :
\[\lambda = 4 \cos ^4 x - 4 \cos ^2 x + 1 - 2(1 - 2 \cos ^2 x + \cos ^4 x) - 2 \cos ^2 x\]
\[\lambda = 2 \cos ^4 x - 2 \cos ^2 x - 1\]
We can factor out a 2 and rewrite this as :
\[\lambda = 2(\cos ^4 x - \cos ^2 x - \frac{1}{2})\]
\[\lambda = 2[(\cos ^2 x - \frac{1}{2})^2 - \frac{3}{4}]\]
So λmax=2[41−43]=2×(4−2)=−1 (maximum value) and
λmin=2[0−43]=−23( minimum value )
So range of the value of x is [2−3,−1].