how to proof cos square + sin square =1?

Answer 1

Start with pythagoras' theorem

opp^2 + adj^2 = hyp^2

Divide throughout by hyp^2 to get
sin^2 + cos^2 = 1

Answer 2

Draw a circle of radius 1. Let P be a point on the circle and O the origin. the angle OP makes with positive x axis, call it a.
The co-ordinates of P are (cos a, sin a). Now use the pythogorean theorem.
hypotenuse ^2 = x^2 + y ^ 2 = cos^2 a + sin ^ a.
but we have a unit circle, so hypotenuse = 1
substitute and get the desired result.

Answer 3

Use a theoretical right angle triangle of length x, height y and hypotenuse h.

Now we know that a^2 + b^2 = c^2
therefore x^2 + y^2 = h^2

now x = h cos(theta)
y = h sin(theta)

h^2 cos^2(theta) + h^2 sin^2(theta) = h^2

divide by h^2

cos^2(theta) + sin^2(theta) = 1

and thus proved!!

Answer 4

Pythagorean Theorem.

That is just how sin and cosine are defined. Use the unit circle.

Answer 5

Consider cos Ø and sin Ø where:-
cos Ø = x / r
sin Ø = y / r
cos² Ø + sin² Ø = x² / r² + y² / r²
cos² Ø + sin² Ø = (x² + y²) / r² = r² / r² = 1