Question

plss solve it......​

Answer 1

Given :

Sin⁴θ - Cos⁴θ = 1-2 cos²θ

To prove :

LHS = RHS

Proof :

RHS = 1-2 cos²θ

LHS = Sin⁴θ - Cos⁴θ

➝ Sin⁴θ - Cos⁴θ = (Sin²θ)² - (Cos²θ)²

➝ (Sin²θ)² - (Cos²θ)² = (Sin²θ - Cos²θ)(Sin²θ + Cos²θ)

We know that :- Sin²θ + Cos²θ = 1

➝ (Sin²θ - Cos²θ) (1)

➝ Sin²θ - Cos²θ

We know that :- Sin²θ = 1- Cos²θ

➝ ( 1- Cos²θ) - Cos²θ

➝ 1- Cos²θ - Cos²θ

➝ 1- 2Cos²θ = RHS

Therefore, LHS = RHS

hence proved

Answer 2

$\bf \underline {We \: have \: to \: prove :-}$

${sin}^{4} θ - {cos}^{4} θ = 1 - 2 {cos}^{2} θ$

We know that,

${sin}^{2}θ + {cos}^{2} θ = 1$

$⇒ {sin}^{2}θ = 1 - {cos}^{2} θ$

$L.H.S = {sin}^{4} θ - {cos}^{4} θ$

$= ( {sin}^{2} θ)^{2} - {( {cos}^{2}θ) }^{2}$

$= ( {sin}^{2}θ + {cos}^{2}θ )( {sin}^{2}θ - {cos}^{2} θ)$

$= 1 \times ( {sin}^{2} θ - {cos}^{2} θ)$

$= {sin}^{2} θ - {cos}^{2} θ$

$= 1 - {cos}^{2} θ - {cos}^{2} θ$

$= 1 - 2 {cos}^{2} θ$

$= R.H.S$

$\therefore{L.H.S = R.H.S}$

[Hence Proved]