Question

9. sin4 theta + 2sin2 theta cos2 theta + cos4 theta =1.​

Answer 1

ANSWER:

To Prove:

• sin^4 ө + 2sin^2 ө cos^2 ө + cos^4 ө = 1

Proof:

We need to prove that,

$\implies\sin^4\theta+2\sin^2\theta\cos^2\theta+\cos^4\theta=1$

Taking and solving LHS,

$\implies\sin^4\theta+2\sin^2\theta\cos^2\theta+\cos^4\theta$

So,

$\implies(\sin^2\theta)^2+2(\sin^2\theta)(\cos^2\theta)+(\cos^2\theta)^2$

We know that,

⇒ a² + 2ab + b² = (a + b)²

Here,

• a = sin²ө
• b = cos²ө

So,

$\implies(\sin^2\theta)^2+2(\sin^2\theta)(\cos^2\theta)+(\cos^2\theta)^2$

$\implies(\sin^2\theta+\cos^2\theta)^2$

But, we know that,

⇒ sin²ө + cos²ө = 1

So,

$\implies(\sin^2\theta+\cos^2\theta)^2$

$\implies(1)^2$

$\bf\implies1=RHS$

HENCE PROVED!!!!