The identity is derived from the Pythagorean theorem applied to the unit circle, where $\sin(x)$ and $\cos(x)$ represent the y and x coordinates, respectively, and the radius is 1.

Divide the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ by $\cos^2(x)$ to obtain $\tan^2(x) + 1 = \sec^2(x)$.

Divide the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ by $\sin^2(x)$ to obtain $1 + \cot^2(x) = \csc^2(x)$.

They allow us to find trigonometric values of angles that are sums or differences of known angles.

They allow us to find trigonometric values of angles that are twice the size of a known angle.

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: $-2\sin(a)\sin(b)$.

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

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

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

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

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

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

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

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

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

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