The minimum value of 4$$Sin^2$$​θ+5$$Cos^2$$​θ is:

Correct option is D

Given 

​$$4Sin^2\theta + 5Cos^2 \theta$$

Concept used:

Use the identity:

​$$sin⁡^2+cos⁡^2=1$$

Solution:

we can express one trigonometric function in terms of the other.

x= $$sin^2 \theta$$ Then, we can write $$Cos^2\theta$$ = 1- x

​substitute into the expression:

​$$4Sin^2\theta + 5Cos^2 \theta$$ = 4x + 5 (1-x)

​4x + 5 (1-x) 

4x + 5 - 5x

-x +5

So the expression becomes-

​$$4Sin^2\theta + 5Cos^2 \theta$$ = -x + 5

​Minimize the expression

Minimize −x+5 The value of x (i.e.,$$\sin^2 \theta$$​) ranges from 0 to 1

When x=0 ,

−x+5= −0+5 = 5

When x=1

−x+5 = −1+5 = 4

Thus, the minimum value occurs when x = 1 and the minimum value of the expression is;

4