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Given the right triangle shown below, prove cos2(θ) + sin2(θ) = 1.

A triangle with sides x, y, and hypotenuse r. Angle theta is opposite to side y.

By
, cos(θ) = StartFraction x Over r EndFraction, and by
, sin(θ) = StartFraction y Over r EndFraction.

Multiplying both sides of the above equations by r, we get that x = r cos(θ) and y = r sin(θ).

The
states that x2 + y2 = r2.

By
, we have [r cos(θ)]2 + [r sin(θ)]2 = r2.

Applying the
, the equation can be written as r2 cos2(θ) + r2 sin2(θ) = r2.

Dividing both sides of the equation by r2 results in cos2(θ) + sin2(θ) = 1.

Given the right triangle shown below prove cos2θ sin2θ 1 A triangle with sides x y and hypotenuse r Angle theta is opposite to side y By cosθ StartFraction x Ov class=

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The blanks are filled by

Laws of Cosine,

Laws of Sine,

Pythagorean Theorem,

Substitution method,

Laws of Exponents

respectively.

What are the Laws of Cosine?

A triangle with sides x, y, and hypotenuse r.

Angle θ is opposite to side y.

[tex]cos(\theta)=\frac{\mbox{adjacent side}}{\mbox{hypotenuse}}=\frac{x}{r}[/tex]  

What are the Laws of Sine?

A triangle with sides x, y, and hypotenuse r.

Angle θ is opposite to side y.

[tex]sin(\theta)=\frac{\mbox{opposite side}}{\mbox{hypotenuse}}[/tex]

What is Pythagorean Identity?

A triangle with sides x, y, and hypotenuse r.

Angle θ is opposite to side y.

Then the property

x²+y²=r²

is called the Pythagorean Identity.

How to fill the blanks?

Given the right triangle shown below, prove cos²(θ) + sin²(θ) = 1.

A triangle with sides x, y, and hypotenuse r.

Angle θ is opposite to side y.

By Laws of Cosine

, cos(θ) = x/r, and

by Laws of Sine

, sin(θ) = y/r.

Multiplying both sides of the above equations by r, we get that

x = r cos(θ) and

y = r sin(θ).

The Pythagorean Theorem states that x² + y² = r².

By Substitution method , we have [r cos(θ)]² + [r sin(θ)]²= r².

Applying the Laws of Exponents, the equation can be written as r² cos²(θ) + r² sin²(θ) = r².

Dividing both sides of the equation by r² results in cos²(θ) + sin²(θ) = 1.

Hence, the blanks are filled by

Laws of Cosine,

Laws of Sine,

Pythagorean Theorem,

Substitution method,

Laws of Exponents

respectively.

To learn more about Pythagorean Identity visit- https://brainly.com/question/15190643

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