1. Factor out the 2: $2(\sin^2(x) + \cos^2(x)) - 1$. 2. Apply the Pythagorean identity: $2(1) - 1$. 3. Simplify: $2 - 1 = 1$.

Express 75 as 45 + 30. Use the sine sum identity: $\sin(45 + 30) = \sin(45)\cos(30) + \cos(45)\sin(30)$. Evaluate.

1. Use the Pythagorean identity: $\cos^2(x) = 1 - \sin^2(x)$. 2. Substitute: $\cos^2(x) = 1 - (\frac{3}{5})^2 = \frac{16}{25}$. 3. Solve for $\cos(x)$: $\cos(x) = \frac{4}{5}$ (positive since x is in the first quadrant).

1. Use the double-angle identity: $\sin(2x) = 2\sin(x)\cos(x)$. 2. Substitute: $\sin(2x) = 2(0.6)(0.8)$. 3. Calculate: $\sin(2x) = 0.96$.

1. Expand using sum and difference identities: $[cos(a)\cos(b) - \sin(a)\sin(b)] - [cos(a)\cos(b) + \sin(a)\sin(b)]$. 2. Simplify: $-2\sin(a)\sin(b)$.

$1 + \tan^2(x) = \sec^2(x)$

$1 + \cot^2(x) = \csc^2(x)$

$\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$

$\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$

$\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$

$\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)$

$\sin(2a) = 2\sin(a)\cos(a)$

$\cos(2a) = \cos^2(a) - \sin^2(a)$

$\cos(2a) = 1 - 2\sin^2(a)$

$\cos(2a) = 2\cos^2(a) - 1$

The fundamental trigonometric identity: $\sin^2(x) + \cos^2(x) = 1$.

$\tan(x) = \frac{\sin(x)}{\cos(x)}$

$\sec(x) = \frac{1}{\cos(x)}$

$\cot(x) = \frac{\cos(x)}{\sin(x)}$

$\csc(x) = \frac{1}{\sin(x)}$