UPSC NDA Math Model Paper 7

y = f(x) = cos2x - sin2x
Multiplying and dividing the above equation with √2
∴y=

√2

√2

(cos‌2x‌−‌sin‌2x)
∴y=√2(cos‌45‌cos‌2x‌−‌sin‌45‌sin‌2x)
∴y=√2{cos(

π

4

‌+‌2x)}=f(x)
We know that the value of trigonometric function cos x and sin x lies between -1 to +1
∴ -1 ≤ cos x ≤ 1
∴−1≤{cos(

π

4

‌+‌2x)}≤1
∴−√2≤‌√2{cos(

π

4

‌+‌2x)} ≤√2
∴−√2≤f(x)≤√2
∴ the range of the function is [−√2,√2].