Question

if (f) = 3x^4 + x^2 + 5 - 3cosx + 2sin^2x then show that f(x) + f(-x) = 2f(x)​

Answer 1

Step-by-step explanation:

Given:-

f(x)=3x^4 + x^2 + 5 - 3cosx + 2sin^2x

To find:-

Show that f(x) + f(-x) = 2f(x)

Solution:-

Given that

f(x)=3x^4 + x^2 + 5 - 3cosx + 2sin^2x

Put x = - x then

f(-x)=

3(-x)^4+(-x)^2+5-3cos(-x)+2sin(-x)^2

=> 3x^4+x^2-3cosx+2(-sin x)^2

Since , Cos(- θ) = Cos θ

Sin(- θ) = - Sin θ

=> 3x^4+x^2-3Cosx +2 sin^2x

f(-x)=3x^4+x^2-3Cosx +2 sin^2x

Now ,

LHS:

f(x)+f(-x)

=> 3x^4+x^2-3Cosx +2 sin^2x + 3x^4+x^2-3Cosx +2 sin^2x

=> (3x^4+3x^4)+(x^2+x^2)+(-3Cos x -3 Cos x ) +

( 2 Sin^2 x + 2 Sin^2 x)

=> 6x^4+2x^2 -6 Cos x +4 sin^2 x

=> 2(3x^4+x^2-3 Cos x +2 Sin^2 x)

=> 2 f(x)

=> RHS

LHS = RHS

f(x)+f(-x) = 2 f(x)

Hence, Proved

Used formulae:-

• Cos(- θ) = Cos θ
• Sin(- θ) = - Sin θ