State the principal value of $\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$,[1] the values between wh

Question

State the principal value of $\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$,[1] the values between which the principal value of $\tan^{-1}x$ must lie,[1] the range of values of $k$ for which the equation $4\sin^2 x = k$ has a solution.[1] Prove the identity $\cot x - \cot x \tan^2 x + \tan x = \frac{1 + \cos 2x}{\sin 2x}$.[4] Hence solve the equation $\cot x - \cot x \tan^2 x + \tan x = \text{cosec}\,2x - 3$ for $0^\circ \le x \le 180^\circ$.[5]

Accepted Answer

(a)(i) $-60^\circ$ (or $-\pi/3$). (ii) $-90^\circ<\tan^{-1}x<90^\circ$. (iii) $0\le k\le4$. (b)(i) Identity proven. (ii) $x=80.8^\circ,\ 170.8^\circ$.

2025 PHS P2 Q9

2025 PHS P2 Q9

Solution:

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