# Prove cos⁴A + sin⁴A = 1 - 1/2(sin²2A)

Can someone please help me to solve this question?

I can't go beyond this:

= cos⁴A + sin⁴A

= (cos^{2}A)^{2} + (sin^{2}A)^{2}

I can't solve it after this :(

Go to main content# Prove cos⁴A + sin⁴A = 1 - 1/2(sin²2A)

Can someone please help me to solve this question?

I can't go beyond this:

= cos⁴A + sin⁴A

= (cos^{2}A)^{2} + (sin^{2}A)^{2}

I can't solve it after this :(

Hi Rames! In this question we need to know:

- a
^{2}+ b^{2}= (a + b)^{2}- 2ab or (a - b)^{2}+ 2ab - cos
^{2}A + sin^{2}A = 1 - 2sinA.cosA = sin2A

Here is the solution:

= cos⁴A + sin⁴A

= (cos^{2}A)^{2} + (sin^{2}A)^{2}

= (cos^{2}A + sin^{2}A)^{2} - 2cos^{2}A.sin^{2}A

= 1 - $\frac{2^2}{2}$ sin^{2}A.cos^{2}A

= 1 - $\frac 12$(2sinA.cosA)^{2}

∴ 1 - $\frac 12$sin^{2}2A = RHS

Here is the video explanation: