Question

If \[2{\cos ^2}\theta - 2{\sin ^2}\theta = 1\] then \[\theta = \]
\[\left( 1 \right){\text{ }}15^\circ \]
\[\left( 2 \right){\text{ 30}}^\circ \]
\[\left( 3 \right){\text{ 45}}^\circ \]
\[\left( 4 \right){\text{ 60}}^\circ \]

Answer 1

Hint: We have to find out the value of theta from the given equation in the question. So our first step is to take out the two common and then we observe that the term inside the bracket is the value of \[\cos \left( {2x} \right)\] .Hence by using this formula we are able to find the value of theta.

Complete step by step solution:
It is given to us that \[2{\cos ^2}\theta - 2{\sin ^2}\theta = 1\]
By taking \[2\] common from the above equation we get
\[2\left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right) = 1\] -------- (i)
In general we know that \[\cos \left( {2x} \right) = {\cos ^2}x - {\sin ^2}x\] .Therefore, the equation (i) becomes
\[2\left( {\cos 2\theta } \right) = 1\]
Now by shifting the number outside the bracket to the right hand side we get
\[\cos 2\theta = \dfrac{1}{2}\]
We all know that if the value of \[\theta \] is \[{\text{60}}^\circ \] that is \[\dfrac{\pi }{3}\] then the value of \[\cos \theta \] is equal to \[\dfrac{1}{2}\] .Therefore,
\[\cos \left( {2\theta } \right) = \cos \left( {\dfrac{\pi }{3}} \right)\]
Both the cos on each side will cancel out each other as if we shift one of the cos to the other side then they are in the division form. Like if we are shifting the right hand cos to the left hand side then it is in the form
\[\dfrac{{\cos \left( {2\theta } \right)}}{{\cos }} = \dfrac{\pi }{3}\] or we can write it as \[{\cos ^{ - 1}}\left( {\cos \left( {2\theta } \right)} \right) = \dfrac{\pi }{3}\] .Also we know that \[{\cos ^{ - 1}}\left( {\cos x} \right) = x\]
Therefore ,
\[2\theta = \dfrac{\pi }{3}\]
Now again by shifting \[2\] to the right hand side we get
\[\theta = \dfrac{\pi }{{3 \times 2}}\]
Or we can write it as
\[\theta = \dfrac{\pi }{6}\] that is \[\theta = 30^\circ \]
Therefore, the required value of \[\theta \] is \[30^\circ \] .
Hence the correct option is \[\left( 2 \right){\text{ 30}}^\circ \].

Note:
The solution to the question will be easier for you if you remember the trigonometric values and the trigonometric formulas. Because they both play a very important role in the trigonometric problems. Keep in mind the trigonometric values in the form of both radian and the degrees.