The simplified value of sin4α+cos4α+12sin22α is …….

Correct option is D

Given:

Expression:
​$$\sin^4\alpha + \cos^4\alpha + 12\sin^2(2\alpha)$$​

Formula Used:

1. $$\sin^4\alpha + \cos^4\alpha = (\sin^2\alpha + \cos^2\alpha)^2 - 2\sin^2\alpha\cos^2\alpha$$​
2. $$\sin^2(2\alpha) = 4\sin^2\alpha\cos^2\alpha$$​

Solution:

1. Simplify $$\sin^4\alpha + \cos^4\alpha$$​ :

​$$\sin^4\alpha + \cos^4\alpha = (\sin^2\alpha + \cos^2\alpha)^2 - 2\sin^2\alpha\cos^2\alpha Since \sin^2\alpha + \cos^2\alpha = 1 : \sin^4\alpha + \cos^4\alpha = 1 - 2\sin^2\alpha\cos^2\alpha$$​

2. Simplify $$12\sin^2(2\alpha)$$​ :

Using $$\sin^2(2\alpha) = 4\sin^2\alpha\cos^2\alpha : 12\sin^2(2\alpha) = 12 \times 4\sin^2\alpha\cos^2\alpha = 48\sin^2\alpha\cos^2\alpha$$​

3. Combine the terms:

​$$\sin^4\alpha + \cos^4\alpha + 12\sin^2(2\alpha) = (1 - 2\sin^2\alpha\cos^2\alpha) + 48\sin^2\alpha\cos^2\alpha Simplify further: \sin^4\alpha + \cos^4\alpha + 12\sin^2(2\alpha) = 1 + 46\sin^2\alpha\cos^2\alpha$$​

4. Final Simplification:

Since $$\sin^2\alpha\cos^2\alpha \leq \frac{1}{4}$$​ , the maximum value of the expression is 1, which is true for all values of $$\alpha$$​ .

Final Answer:

​$$\mathbf{D. \; 1}$$​