Question

ExA
prove the following statements
cos^4 A - sin^4 A+1=2 cos^2 A​

Answer 1

$\huge \red{ \mathfrak{solution}}$

To prove :

$\cos^{4} ( \alpha ) - { \sin }^{4} \alpha + 1 = 2 { \cos }^{{2} } \alpha$

LHS

$\leadsto \: {cos}^{4} \alpha - {sin}^{4} \alpha + 1 \\ \\ \leadsto \: {( {cos}^{2} \alpha })^{2} - {sin}^{4} \alpha + 1 \\ \\ \leadsto \: (1 - {sin}^{2} \alpha ) ^{2} - {sin}^{4} \alpha + 1 \\ \\ \leadsto \: 1 + {sin}^{4} \alpha \: - 2 {sin}^{2} \alpha - {sin}^{4} \alpha + 1 \\ \\ \leadsto \: 2 - 2 {sin}^{2} \alpha \\ \\ \leadsto \: 2(1 - {sin}^{2} \alpha ) \\ \\ \leadsto2 {cos}^{2} \alpha$

LHS = RHS

$\huge \: \boxed{ \mathfrak{proved}}$

Answer 2

Solution:

Given:

➜ Prove the following statements

cos^4 A - sin^4 A+1=2 cos^2 A

To Prove:

➜ cos² (a) - sin⁴ a + 1

LHS:

➜ cos² a - sin⁴ a + 1

➜ (cos² a)² - sin⁴ a + 1

➜ (1 - sin² a) ² - sin² a + 1

➜ 1 + sin⁴ a - 2sin² a - sin⁴ a + 1

➜ 2 - 2 sin² a

➜ 2(1 - sin² a)

➜ 2 cos² a

LHS = RHS

Know Terms:

• LHS => Left hand side.
• RHS => Right hand side.

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