Question

trigonometric double angle identities verifying an identity the following steps show how the right side of ( 1 + sin(2x) = sin(x) + cos(x)^2 ) can be rewritten to show it is an identity. ( sin(x) + cos(x)^2 ) ( = sin^2(x) + 2sin(x)cos(x) + cos^2(x) ) ( = 1 + 2sin(x)cos(x) ) ( = 1 + sin(2x) ) what are the correct justifications, listed in proper order? options (from boxes, top to bottom, left to right): distributive property, double angle identity, pythagorean identity; distributive property, pythagorean identity, double angle identity; double angle identity, distributive property, pythagorean identity; pythagorean identity, distributive property, double angle identity

Explanation:

To solve this, we analyze each step of the identity verification:

Step 1: Expand \([\sin(x) + \cos(x)]^2\)

This uses the Distributive Property (specifically, the formula \((a + b)^2 = a^2 + 2ab + b^2\), which is an application of the distributive property for squaring a binomial).

Step 2: Simplify \(\sin^2(x) + \cos^2(x)\)

The Pythagorean Identity states \(\sin^2(\theta) + \cos^2(\theta) = 1\), so \(\sin^2(x) + \cos^2(x) = 1\). This uses the Pythagorean Identity.

Step 3: Simplify \(1 + 2\sin(x)\cos(x)\)

The Double Angle Identity for sine is \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\), so \(2\sin(x)\cos(x) = \sin(2x)\). This uses the Double Angle Identity.

The correct order of justifications is: Distributive Property, Pythagorean Identity, Double Angle Identity.

Answer:

Distributive Property, Pythagorean Identity, Double Angle Identity (matching the option with this order).