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# If $\sin A + {\left( {\sin A} \right)^2} = 1$, then the value of the expression $\left[ {{{\left( {\cos A} \right)}^2} + {{\left( {\cos A} \right)}^4}} \right]$ is${\text{A}}{\text{. 1}} \\ {\text{B}}{\text{. }}\dfrac{1}{2} \\ {\text{C}}{\text{. 2}} \\ {\text{D}}{\text{. 3}} \\$

Last updated date: 15th Sep 2024
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Hint: Here, we will be using the formula ${\left( {\sin \theta } \right)^2} + {\left( {\cos \theta } \right)^2} = 1$ in order to determine the values of ${\left( {\cos A} \right)^2}$ and ${\left( {\cos A} \right)^4}$ from the given equation which is $\sin A + {\left( {\sin A} \right)^2} = 1$ and then ultimately the expression whose value is required will appear as the LHS of the given equation.

Given, $\sin A + {\left( {\sin A} \right)^2} = 1{\text{ }} \to {\text{(1)}} \\ \Rightarrow \sin A = 1 - {\left( {\sin A} \right)^2}{\text{ }} \to {\text{(2)}} \\$
As we know that
${\left( {\sin \theta } \right)^2} + {\left( {\cos \theta } \right)^2} = 1 \\ \Rightarrow {\left( {\cos \theta } \right)^2} = 1 - {\left( {\sin \theta } \right)^2}{\text{ }} \to {\text{(3)}} \\$
Replacing the angle $\theta$ with angle $A$ in equation (3), we get
$\Rightarrow {\left( {\cos A} \right)^2} = 1 - {\left( {\sin A} \right)^2}{\text{ }} \to {\text{(4)}}$
Clearly, the RHS of both the equations (2) and (4) are the same so the LHS of both the equations will also be equal.
$\Rightarrow \sin A = {\left( {\cos A} \right)^2}{\text{ }} \to {\text{(5)}}$
So, the value of the expression $\left[ {{{\left( {\cos A} \right)}^2} + {{\left( {\cos A} \right)}^4}} \right]$ can be determined by little modification as under.
$\left[ {{{\left( {\cos A} \right)}^2} + {{\left( {\cos A} \right)}^4}} \right] = \left[ {{{\left( {\cos A} \right)}^2} + {{\left( {\cos A} \right)}^2} \times {{\left( {\cos A} \right)}^2}} \right]$
Using equation (5), we get
$\Rightarrow \left[ {{{\left( {\cos A} \right)}^2} + {{\left( {\cos A} \right)}^4}} \right] = \left[ {\sin A + \left( {\sin A} \right) \times \left( {\sin A} \right)} \right] \\ \Rightarrow \left[ {{{\left( {\cos A} \right)}^2} + {{\left( {\cos A} \right)}^4}} \right] = \left[ {\sin A + {{\left( {\sin A} \right)}^2}} \right] \\$
Finally using the given equation (1), we get
$\Rightarrow \left[ {{{\left( {\cos A} \right)}^2} + {{\left( {\cos A} \right)}^4}} \right] = 1$
Therefore, the value of the expression $\left[ {{{\left( {\cos A} \right)}^2} + {{\left( {\cos A} \right)}^4}} \right]$ is 1.

Hence, option A is correct.

Note: In this particular problem, we obtained the value of ${\left( {\cos A} \right)^2}$ in terms of $\sin A$using the given equation and some trigonometric formula. From there we represented the expression whose value is required in terms of ${\left( {\cos A} \right)^2}$ which is ultimately converted in terms of $\sin A$.