Trigonometric Ratios Question 2
Question 2 - 29 January - Shift 2
The set of all values of $\lambda$ for which the equation $\cos ^{2} 2 x-2 \sin ^{4} x-2 \cos ^{2} x=\lambda$
(1) $[-2,-1]$
(2) $[-2,-\frac{3}{2}]$
(3) $[-1,-\frac{1}{2}]$
(4) $[-\frac{3}{2},-1]$
Show Answer
Answer: (4)
Solution:
Formula: Pythagorean Identities, Double Angle Identities, Range of Trigonometric Functions
$\lambda=\cos ^{2} 2 x-2 \sin ^{4} x-2 \cos ^{2} x$
convert all in to $\cos x$.
$ \begin{aligned} \lambda & =(2 \cos ^{2} x-1)^{2}-2(1-\cos ^{2} x)^{2}-2 \cos ^{2} x \\ & =4 \cos ^{4} x-4 \cos ^{2} x+1-2(1-2 \cos ^{2} x+\cos ^{4} x)- 2 \cos ^{2} x \\ & =2 \cos ^{4} x-2 \cos ^{2} x+1-2 \\ & =2 \cos ^{4} x-2 \cos ^{2} x-1 \\ & =2[\cos ^{4} x-\cos ^{2} x-\frac{1}{2}] \\ & =2[(\cos ^{2} x-\frac{1}{2})^{2}-\frac{3}{4}] \\ \lambda _{\max } & =2[\frac{1}{4}-\frac{3}{4}]=2 \times(-\frac{2}{4})=-1 \text{ (max Value) } \\ \lambda _{\min } & =2[0-\frac{3}{4}]=-\frac{3}{2}(\text{ Minimum Value) } \end{aligned} $
So, Range $=[-\frac{3}{2},-1]$
