Question

cos^4 A + sin^4A + 2sin²A cos²A = 1

Answer 1

cos^4 a - sin^4 a = cos2a

Let us rewrite:

(cos^2 a)^2 - (sin^2 a)^2

We kno wthat:

a^2 - b^2 = (a-b)(a+b)

==> cos^4 a- sin^4 a=(cos^2 a - sin^2 a)(cos^2 a + sin^2 a)

Also:

We know that:

cos^2 a-sin^2 a = cos2a

cos^2 a + sin^2 a= 1

Now substitute:

cos^4 a - sin^4 a= cos2a * 1

                              = cos2a

==> the equality is true.



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Answer 2

$\cos^4(A) + \sin^4(A) + 2 \sin^2(A)\cos^2(A) = 1$

$(\cos^2(A))^2 + (\sin^2(A))^2 + 2 \sin^2(A)\cos^2(A) = 1$

$(\cos^2(A)+ \sin^2(A))^2 = 1$ Using formula $a^2+b^2+2ab=(a+b)^2$

$(\cos^2(A)+ \sin^2(A))^2 = 1$

Now use formula $(\cos^2(A)+ \sin^2(A))^2 = 1$

so we get 1=1 which is true.

Hence given equation is true.