Question

Use trigonometric identities to transform the left side of the equation into the right side (0 < θ < π/2).

sin2 θ − cos2 θ = sin2 θ − (_______)
= 2 sin2 θ − 1

Answer 1

The blank should be filled with the trigonometric identity “cos2 θ = 1 − sin2 θ”, which when substituted gives the right side of the equation as 2 sin2 θ − 1.

Step-by-step explanation:

The question involves using trigonometric identities to simplify or transform expressions. Given the expression sin2 θ − cos2 θ, we want to transform it into the form sin2 θ − (_______) = 2 sin2 θ − 1. The blank should be filled with an expression equivalent to cos2 θ that will lead to the desired form.

To transform the left side into the right side, use the Pythagorean identity cos2 θ = 1 − sin2 θ. Inserting this identity, we get:

sin2 θ − cos2 θ = sin2 θ − (1 − sin2 θ)

Simplifying the expression gives us:

sin2 θ − 1 + sin2 θ = 2 sin2 θ − 1

Thus, the transformed equation is equivalent to the given right side of the equation.

Answer 2

To transform the left side of the equation into the right side under the interval (0 < θ < π/2), the missing trigonometric identity to complete the equation is cos2 θ = 1- sin2 θ. This is a trigonometric identity specifically under the Pythagorean Identities