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Firstly, sin2(x) + cos2(x) = 1 is true for ANY value of x. It's a very important identity to know in trigonometry. (note: every use of "2" in this response means "to the power of 2")

To prove this fact, one can assume the Pythagorean Theorem which states that given a right triangle, opp2+adj2=hyp2, where opposite and adjacent are the two smaller sides and the hypoteneuse is the diagonal of the triangle.

It is also important to note that sin(x)=opp/hyp and cos(x)=adj/hyp.

Okay, so to begin, lets assume that the equation opp2+adj2=hyp2 is true.

Now divide both sides algebraically by hyp2

This yields the equation (opp2+adj2)/hyp2=1

Now distribute over the parenthesis to get opp2/hyp2+adj2/hyp2=1

Lastly, simplify the exponents: (opp/hyp)2+(adj/hyp)2=1

Since sin(x)=opp/hyp and cos(x)=adj/hyp, these can be substituted above:

sin2(x)+cos2(x)=1

Since this equation holds for any value of x, it must hold for x=pi/4

Sin(π/4) = 1/√2

Cos(π/4) = 1/√2

=>Sin²(π/4) = (1/√2)² = 1/2

=> Cos²(π/4) = (1/√2)² = 1/2

1/2 + 1/2 = 1

Usually when you are given a specific value, then that is all you need to use to make your 'proof'. Otherwise you have to prove it in general with all the math in all its glory.

not only when Θ=π/4, sin²Θ+cos²Θ=1 is true with any Θ.

Draw a right triangle then use Pythagoras rule to prove it...

Just substitute the value of Θ. Since the value of sinπ/4=1/√2and cosπ/4=1/√2, if you square the value of sinπ/4 and cosπ/4 their sum would be equal to 1.