If \[\cos {40^\circ } = x\] and\[\cos \theta = 1 - 2{x^2}\], then the possible values of \[\theta \] lying between \[{0^\circ }\]and \[{360^\circ }\]is
A. \[{100^\circ }\]and\[{260^\circ }\]
B. \[{80^\circ }\]and \[{280^\circ }\]
C. \[{280^\circ }\]and \[{110^\circ }\]
D. \[{110^\circ }\]and\[{260^\circ }\]

Hint: A trigonometric equation's general equation is the answer that includes all feasible solutions. Simply said, trigonometric equations are equations that include the sine and cosine trigonometric ratios on the variable. In this case, for the equation \[\cos {40^\circ } = x\]substitute the value of \[\cos \theta = 1 - 2{x^2}\] and take common variables outside and solve.





Complete step by step solution:We have given that the equation is \[\cos {40^\circ } = x\] and\[\cos \theta = 1 - 2{x^2}\]
From the given equation, as per the question;
Substitute the value of \[\cos {40^\circ } = x\] in \[\cos \theta = 1 - 2{x^2}\]
Observe the corresponding \[ = - \left( {2{{\cos }^2}{{40}^\circ } - 1} \right)\] value from the question.
\[ \Rightarrow \cos \theta = 1 - 2{\cos ^2}{40^\circ }\]
Take the minus sign from the above equation as common:
Multiply the minus sign with the term inside the parentheses:
\[ = - 2{\cos ^2}\left( {{{40}{\circ \:}}} \right) + 1\]
From the equation obtained, take\[\cos \]as common:
\[ = - \cos \left( {2 \times {{40}^\circ }} \right)\]
Multiply the term inside the parentheses to obtain the less complicated term:
\[ = - \cos {80^\circ }\]
Therefore, we will obtain two forms of equations as below:
\[\cos \theta = \cos \left( {{{180}^\circ } + {{80}^\circ }} \right)\]
Or
\[\cos \theta = \left( {\cos {{180}^\circ } - {{80}^\circ }} \right)\]
Hence, the possible values of \[\theta \]lying between \[{0^\circ }\]and\[{360^\circ }\]is \[\theta = {100^\circ }\]and\[{260^\circ }\]



Option ‘A’ is correct

Note: ometimes students make error in applying identities and are not able to comprehend how to arrive at the other. Students sometimes apply identities incorrectly\[sin{\rm{ }}\theta {\rm{ }} = {\rm{ }}1 - cos{\rm{ }}\theta \] \[tan{\rm{ }}\theta = {\rm{ }}1 + {\rm{ }}sec{\rm{ }}\theta \]. Starting the simplification of the either sides simultaneously are the common mistakes made while solving using trigonometry identities. Like terms should be grouped at the same side, so as to simplify the equation in much easier way. One thing we must remember is that all trigonometric ratios have the property that sin is not the same as sin because it denotes a ratio rather than a product.