COMEDK 2024 Shift 1 Paper

We can start by squaring both sides of the equation cos‌θ=‌

1

2

(x+‌

1

x

) :

cos2θ=‌

1

4

(x+‌

1

x

)2
Expanding the right side gives:
cos2θ=‌

1

4

(x2+2+‌

1

x2

)
Multiplying both sides by 4 :
4cos2θ=x2+2+‌

1

x2

Subtracting 2 from both sides:
4cos2θ−2=x2+‌

1

x2

Finally, dividing both sides by 2 :
2cos2θ−1=‌

1

2

(x2+‌

1

x2

)

Using the trigonometric identity cos2θ+sin‌2θ=1, we can rewrite 2cos2θ−1 as:
2cos2θ−(cos2θ+sin‌2θ)=cos2θ−sin‌2θ
Therefore, ‌

1

2

(x2+‌

1

x2

)=cos2θ−sin‌2θ
The correct answer is Option C: cos2θ−sin‌2θ