Question

prove sin^4a+2sin^2acos^2a+cos^4a=1​

Answer 1

Step-by-step explanation:

[sin^2(a) + cos^2(a)]^2

= 1

Answer 2

Take the LHS.

sin⁴a + 2 sin²a cos²a + cos⁴a

This seems in the form x² + 2xy + y², where x = sin²a and y = cos²a.

This is the expanded form of (x + y)², isn't it?

So we get,

sin⁴a + 2 sin²a cos²a + cos⁴a = (sin²a + cos²a)².

We are very much familiar with this one.

sin²a + cos²a = 1.

This is because,

    sin²a + cos²a

⇒  (opposite side of angle a / hypotenuse)² + (adjacent side of angle a / hypotenuse)²

⇒  ((opposite side of angle a)² + (adjacent side of angle a²)) / hypotenuse²

⇒  hypotenuse² / hypotenuse²   [Right triangle is considered, so base² +  altitude² = hypotenuse²]

⇒  1

So,

(sin²a + cos²a)² = 1² = 1 which is the RHS.

Hence Proved!