The identity sin²θ + cos²θ = 1 is one of the fundamental trigonometric identities derived from the Pythagorean theorem. The other options mix different functions or represent other ratios and do not express this basic identity.

By applying the Pythagorean identity, sin²θ + cos²θ simplifies directly to 1. The other options do not utilize this fundamental relationship.

Cosecant is defined as the reciprocal of sine, so cscθ = 1/sinθ. The other options represent different trigonometric relationships.

Since sin²θ + cos²θ = 1, subtracting cos²θ from 1 yields sin²θ. This direct application of the Pythagorean identity is the correct simplification.

Tangent and cotangent are reciprocal functions, meaning tanθ · cotθ = 1. Therefore, cotθ is correctly identified as the reciprocal of tangent.

The double-angle formula for sine is given by sin2θ = 2 sinθ cosθ, which expresses the sine of twice an angle in terms of sine and cosine of the original angle. The other options correspond either to cosine double-angle formulas or incorrect expressions.

The identity cos2θ = cos²θ - sin²θ is a standard form of the double-angle formula for cosine. The other options either simplify to 1, represent the sine double-angle, or are mathematically incorrect.

The half-angle formula for sine is sin(θ/2) = ±√((1 - cosθ)/2), with the ± sign depending on the quadrant of the angle. Other options either omit the necessary ± sign or use the incorrect expression inside the square root.

Using the Pythagorean identity sin²θ + cos²θ = 1, and given cosθ = 0.6, sinθ is calculated as √(1 - 0.36) which equals 0.8. Since the angle is in the first quadrant, the positive square root is taken.

Tangent is defined as the ratio of sine to cosine, i.e., tanθ = sinθ/cosθ. The other alternatives do not represent the correct ratio for tangent.

Starting with tanθ = sinθ/cosθ, squaring gives tan²θ = sin²θ/cos²θ. By expressing 1 as cos²θ/cos²θ and combining terms, we get (sin²θ + cos²θ)/cos²θ which simplifies to 1/cos²θ = sec²θ. The incorrect options do not perform this standard manipulation.

Cofunction identities relate the trigonometric functions of complementary angles. The identity sin(90° - θ) = cosθ accurately reflects this relationship, while the other options are either misstatements or incorrect.

The expression 2sinθcosθ is the double-angle formula for sine, which directly simplifies to sin2θ. The other alternatives do not correctly represent this well-known identity.

The Pythagorean identity 1 + tan²θ = sec²θ can be rearranged to give tan²θ = sec²θ - 1. This is the correct transformation, unlike the other provided options which misapply the identity.

The half-angle identity for sine squared indicates that sin²θ = (1 - cos2θ)/2, making this the correct simplification. The remaining responses do not match this standard trigonometric result.

Using the cosine double-angle identity where cos2θ = cos²θ - sin²θ, it follows that sin²θ - cos²θ equals - (cos²θ - sin²θ), which is -cos2θ. The other options incorrectly represent the relationship between these terms.

The double-angle identity cos2θ = 1 - 2sin²θ allows us to directly simplify 1 - 2sin²θ to cos2θ. The other responses do not accurately reflect this standard identity.

Using the sum-to-product formula sinA + sinB = 2 sin((A+B)/2) cos((A-B)/2), with A = θ and B = 3θ results in 2 sin2θ cosθ. The alternatives do not correctly apply the formula.

The correct half-angle formula for cosine is cos(θ/2) = ±√((1 + cosθ)/2), with the sign determined by the quadrant of the angle. The other options either omit the ± sign or substitute the wrong expression inside the square root.

Given sinθ = 0.8, we find cosθ using sin²θ + cos²θ = 1, which gives cosθ = 0.6 in the first quadrant. Then, applying the double-angle formula cos2θ = cos²θ - sin²θ leads to 0.36 - 0.64 = -0.28. The other values result from miscalculations.