# The Maximum value of $ 4{\sin ^2}x + 3{\cos ^2}x $ is

$ A - 3 $

$ B - 4 $

$ C - 5 $

$ D - 7 $

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**Hint**: In order to solve the Trigonometric Numerical , we need to know all the Identities by heart in order to solve the numerical . First step is to identify the type of identity to be used based on the trigonometric function given in the question. Since there are several identities for a single function, the next step is to decide which particular identity would suffice in order to solve the problem. It is also Advisable to Know the Maximum value , Minimum value and value for $ \dfrac{\pi }{6}\& {60^ \circ } $ for all trigonometric functions

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__Complete step-by-step answer__From the above question we can figure it out that we have to make use of the Trigonometry identities in order to find the correct answer. Since the problem involves $ \sin x $ & $ \cos x $ we will have to make use of identity

$ {\sin ^2}\theta + {\cos ^2}\theta = 1..........(1) $

Given : $ 4{\sin ^2}x + 3{\cos ^2}x $

Restructuring $ Equation1 $ :

$ {\cos ^2}\theta = 1 - {\sin ^2}\theta ..........(2) $

Substituting value of \[{\cos ^2}\theta \]from $ Equation2 $ into the given problem

We are replacing $ \theta $ with $ x $ , since the question consists of $ x $ . There is no compulsion on using $ \theta $ with Trigonometric functions.

\[ \Rightarrow 4{\sin ^2}x + 3 \times (1 - {\sin ^2}x)\]

On simplifying the above equation and on opening the brackets we will get the following equation

$ \Rightarrow 4{\sin ^2}x + 3 - 3{\sin ^2}x........(3) $

Further Simplifying the $ Equation3 $ in order to bring it to non-reducible form

$ \Rightarrow {\sin ^2}x + 3.......(4) $

Now remembering the properties of $ \sin x $ we can say that the value of $ \sin x $ ranges from $ - 1 $ to $ 1 $ .

Thus the maximum value of $ \sin x $ will always be $ 1 $ . Thus the maximum value of \[{\sin ^2}x\] will be $ 1 $ .

$ \therefore $ Substituting Maximum Value of $ {\sin ^2}x $ $ Equation4 $

Final answer on substituting would be $ 4 $ .

Thus the maximum value of $ 4{\sin ^2}x + 3{\cos ^2}x $ is $ 4 $ .

From the given options, the correct option is $ OptionB $ which has a value of $ 4 $ .

**So, the correct answer is “Option B”.**

**Note**: It is also Advisable to Know the Maximum value , Minimum value and value $ \dfrac{\pi }{6}\& {60^ \circ } $ for all trigonometric functions. This is because in almost all the sums these are the standard angles based on which the questions are asked. Apart from these , memorizing the formulas and the relationship between various trigonometric functions is extremely vital.