(sin145)^2 = 1-(cos35)^2

Let us rewrite:

sin145 = sin(180-35)

But we know that:

sin 180-a = sin a

==> sin 145 = sin35

Now substitute:

(sin35)^2 = 1- (cos35)^2

==> (sin35)^2 + (cos35)^2 = 1

We know that (sina)^2 + (cosa)^2 = 1

Then the equality is true.

We can write instead of sin 145 = sin 35.

We'll verify if we could make the change, using the following formula:

sin (a-b) = sin a * cos b - sin b* cos a

We'll put a = 180 and b = 35

sin(180-35) = sin 145 = sin 180*cos 35 - sin 35 * cos 180

But sin 180 = 0 and cos 180 = -1

sin(180-35) = 0 - (- sin 35)

sin 145 = sin 35

We'll substitute sin 145 by sin 35 and we'll apply the fundamental formula of trigonometry:

**(sin 35)^2 + (cos 35)^2 = 1**

We'll subtract (cos 35)^2 both sides:

(sin 35)^2 = 1 - (cos 35)^2

But sin 35 = sin 145, so the expression is verified!

To prove that sin(145)^2= 1-(cos35)^2.

Proof:

We know that (sinx)^2+(cosx)^2 = 1 is an identity for all x....(1)

We know that sin (180-x) = sinx. So applying this in the left side of the given equation, we get:

sin 145 = sin(180-145) = sin35. Therefore the given equation becomes:

(sin 35)^2 = 1-(cos35)^2

(sin35)^2+(cos35)^2 = 1 which is true by virtue of the identity at (1)