Question

Choose the CORRECT set of functions, which are linearly dependent.

• A. sin x, $si{n}^{2}xandco{s}^{2}x$
• B. $cosx,sinx,andtanx$
• C. cos 2x, $si{n}^{2}xandco{s}^{2}x$
• D. cos 2x, sin x and cos x

Answer 1

The correct option is C cos 2x, $si{n}^{2}xandco{s}^{2}x$

Let $y_1=cos2x,y_2=co{s}^{2}x,y_3=si{n}^{2}x$

then, $cos2x=co{s}^{2}x-si{n}^{2}x$

$\Rightarrow cos2x=\left(1\right)co{s}^{2}x+(-1)si{n}^{2}x$

$\Rightarrow \left(1\right)y_1-\left(1\right)y_2+\left(1\right)y_3=0$

$\Rightarrow C_1{y}_1+C_2{y}_2+C_3{y}_3=0$

So, linear combination exist between $y_1,y_2,y_3$ So, they are linearly dependent.