Question

The greatest value of ${{\sin }^{4}}\theta +{{\cos }^{4}}\theta$ is
A. $\dfrac{1}{2}$
B. 1
C. 2
D. 3

Answer 1

In order to solve this question, we need the basic trigonometric properties and relations. We also need to know the standard values of the basic trigonometric functions. We need to write the above equation in terms of ${{\left( a+b \right)}^{2}}-2ab$ where a represents ${{\sin }^{2}}\theta$ and b represents ${{\cos }^{2}}\theta .$ We then use the basic relation ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ in order to obtain the answer.

Complete step-by-step answer:
For the given question ${{\sin }^{4}}\theta +{{\cos }^{4}}\theta ,$ let us simplify this equation such that we can solve this easily. Consider the given equation in question,
$\Rightarrow {{\sin }^{4}}\theta +{{\cos }^{4}}\theta$
This above equation can be considered to be of the form ${{a}^{2}}+{{b}^{2}},$ where a represents ${{\sin }^{2}}\theta$ and b represents ${{\cos }^{2}}\theta .$ We know the general expansion of ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab.$ Rearranging the terms,
$\Rightarrow {{a}^{2}}+{{b}^{2}}={{\left( a+b \right)}^{2}}-2ab\ldots \ldots \left( 1 \right)$
Using the relations $a={{\sin }^{2}}\theta$ and $b={{\cos }^{2}}\theta ,$
$\Rightarrow {{\left( {{\sin }^{2}}\theta \right)}^{2}}+{{\left( {{\cos }^{2}}\theta \right)}^{2}}$
Using equation 1,
$\Rightarrow {{\left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)}^{2}}-2{{\sin }^{2}}\theta {{\cos }^{2}}\theta$
We know the basic relation ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1,$ and we use this in the above equation.
$\Rightarrow {{\left( 1 \right)}^{2}}-2{{\sin }^{2}}\theta {{\cos }^{2}}\theta$
We also know that $\sin \left( 2\theta \right)=2\sin \theta \cos \theta ,$ squaring both sides and dividing both sides by 2,
$\Rightarrow \dfrac{1}{2}{{\sin }^{2}}\left( 2\theta \right)=2{{\sin }^{2}}\theta {{\cos }^{2}}\theta$
Substituting this in the above equation,
$\Rightarrow 1-\dfrac{1}{2}{{\sin }^{2}}\left( 2\theta \right)$
We need to find the greatest value of this and this can be found by finding the minimum of the second term. This occurs at $\theta =0.$ Substituting,
$\Rightarrow 1-\dfrac{1}{2}{{\sin }^{2}}\left( 2.0 \right)$
We know that $\sin 0=0.$ Therefore, the greatest value of the above expression is,
$\Rightarrow 1-0=1$
Hence, the greatest value of ${{\sin }^{4}}\theta +{{\cos }^{4}}\theta$ is 1.

So, the correct answer is “Option B”.

Note: It is important to know the basic trigonometric formulae and relations. It is to be noted that since they asked us the maximum value of the above equation, we need to take the minimum value of $\theta .$ If we were to find the minimum value of the given equation, we need to substitute $\theta$ such that ${{\sin }^{2}}\left( 2\theta \right)$ is made maximum.