Question

Prove that cos⁴A + sin⁴A-2sin²A cos²A=1​

Answer 1

I have attached the full solution you may go through it

Answer 2

To prove -

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

1−2sin(a

2

).cos(a)

2

sin(a

4

).cos(a

4

)

=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 -

\begin{gathered} \frac{1 - 2 {sin(a)}^{2}.cos( {a)}^{2} }{1 - 2sin ({a})^{2} .cos ({a})^{2} } \\ = 1\end{gathered}

1−2sin(a)

2

.cos(a)

2

1−2sin(a)

2

.cos(a)

2

=1

→ 1 = LHS= RHS

hence proved.

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