PUCET (Punjab) Logarithms — practice questions

Supplementary-angle identity: $\sin(180° - \theta) = \sin\theta$. Reason: sine is positive in the second quadrant (where $180° - \theta$ lies for $0° < \theta < 180°$), and the reference angle is $\theta$. Companion identities: - $\cos(180° - \theta) = -\cos\theta$ - $\tan(180° - \theta) = -\tan\theta$

The **Pythagorean identity** $\sin^2\theta + \cos^2\theta = 1$ holds for every real $\theta$. It comes from Pythagoras applied to the unit circle: any point on it has coordinates $(\cos\theta, \sin\theta)$ with distance 1 from origin so the sum of squared coordinates is 1. Companion identities: - Dividing by $\cos^2\theta$: $1 + \tan^2\theta = \sec^2\theta$. - Dividing by $\sin^2\theta$: $\cot^2\theta + 1 = \csc^2\theta$.

Angle-sum formula for sine: $\sin(A + B) = \sin A \cos B + \cos A \sin B$. Companion: $\sin(A - B) = \sin A \cos B - \cos A \sin B$ (option C — common trap, the sine of the *difference*). For cosine: $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ (sign flips inside vs outside, option D is $\cos(A+B)$). Sine is not linear in its argument: $\sin(A+B) \neq \sin A + \sin B$ generally.