Question

prove that sin⁴A+cos⁴A/1-2sin²A×cos²A=1​

Answer 1

Step-by-step explanation:

${( { \sin(x)}^{2} + { \cos(x) }^{2} ) }^{2} = { \sin(x) }^{4} + { \cos(x) }^{4} + 2 { \sin(x) }^{2} { \cos(x) }^{2}$

${ \sin(x) }^{2} + { \cos(x) }^{2} = 1$

Using these identities you can prove the relation.

Answer 2

To prove -

$\frac{ \sin( {a}^{4} ). \cos( {a}^{4} ) }{1 - 2 \sin( {a}^{2}) . { \cos(a) }^{2} } = 1$

Proof -

LHS →

Taking Numerator -

sin⁴a + cos ⁴a →( sin²a)² + (cos²a)²

As we know that -

sin²a + cos²a → ( sina +cosa)² - 2sina.cosa.

So value of sin⁴a + cos ⁴a will -

(sin²a+cos²a)- 2sin²a.cos²a

As we know that sin²a+cos²a = 1 , so -

(1)² - 2sin²acos²a

Numerator → 1 - 2cos²a . sin²a

Now putting the value of numerator in the fraction -

$\frac{1 - 2 {sin(a)}^{2}.cos( {a)}^{2} }{1 - 2sin ({a})^{2} .cos ({a})^{2} } \\ = 1$

→ 1 = LHS= RHS

hence proved.