Trigonometric Proof (Cambridge International A-Level Mathematics): Exam Questions

Show that

$\frac{\text{sin}2θ}{2\text{sin}θ} \equiv \text{cos} θ$

where $θ \neq k π$ and $k$ is an integer.

Show that

$\text{sin}^{2} θ ( \text{sec}^{2} θ + \text{cosec}^{2} θ ) \equiv \text{sec}^{2} θ$

(i) Use the quotient rule to show that

\dfrac{\text{d}}{\text{d}x} \left[ \text{cosec} \; x \right] = \dfrac{- \text{cos} \; x}{\text{sin}^2 \; x}

(ii) Hence show that

\dfrac{\text{d}}{\text{d}x} \left[ \text{cosec} \; x \right] = - \text{cot} \; x \; \text{cosec} \; x

Show that

$3 \text{sin} 2 θ - 2 \text{sin} θ \equiv 2 \text{sin} θ ( 3 \text{cos} θ - 1 )$

Show that

$2 \text{cosec} 2 x \text{cot} x \equiv \text{cosec}^{2} x$

where $x \neq \frac{kπ}{2}$ and $k$ is an integer.

Use the identity

$R \text{sin} ( θ + α ) \equiv R \text{cos} α \text{sin} θ + R \text{sin} α \text{cos} θ$

to show that

4 \; \text{sin} \left( \theta + \dfrac{\pi}{4} \right) \equiv 2\sqrt{2} \; (\text{sin} \; \theta + \text{cos} \; \theta)

Given the identity

$\text{sin}^{2} θ + \text{cos}^{2} θ \equiv 1$

show that

(i) $\text{sec}^{2} θ \equiv 1 + \text{tan}^{2} θ$

(ii) $\text{cosec}^{2} θ \equiv 1 + \text{cot}^{2} θ$

Given the identity

$\text{cos} ( A + B ) \equiv \text{cos} A \text{cos} B - \text{sin} A \text{sin} B$

show that

(i) $\text{cos} 2 θ \equiv \text{cos}^{2} θ - \text{sin}^{2} θ$

(ii) $\text{cos} 2 θ \equiv 1 - 2 \text{sin}^{2} θ$

(iii) $\text{cos} 2 θ \equiv 2 \text{cos}^{2} θ - 1$

Show that

$\frac{4\text{sin}^{4}θ}{\text{sin}^{2}2θ} \equiv \text{tan}^{2} θ$

where $θ \neq k π$ and $k$ is an integer.

By using the double angle formula for cosine, show that

$\text{cos} 4 θ \equiv 8 \text{cos}^{4} θ - 8 \text{cos}^{2} θ + 1$

Show that

$\text{sin} θ ( \text{cosec}^{2} θ - 2 ) \equiv \frac{\text{cos}2θ}{\text{sin}θ}$

Show that

$\text{sin} 3 θ + \text{sin} θ \equiv 4 \text{sin} θ - 4 \text{sin}^{3} θ$

Show that

$\frac{4\text{cot}x\text{cos}2x}{\text{sin}4x} \equiv \text{cosec}^{2} x$

where $x \neq \frac{kπ}{4}$ and $k$ is an integer.

Show that

\sqrt{2} \; \text{sin} \left( \theta - \dfrac{\pi}{4} \right) \equiv \text{sin} \; \theta - \text{cos} \; \theta

Show that

$\text{sin} 3 θ \equiv 3 \text{sin} θ - 4 \text{sin}^{3} θ$

Show that

$\text{cos} 4 θ + \text{cos} \frac{π}{3} \equiv 8 \text{sin}^{4} θ - 8 \text{sin}^{2} θ + \frac{3}{2}$

Show that

$\text{tan} 4 θ \equiv \frac{4\text{tan}θ(1-\text{tan}^{2}θ)}{1-6\text{tan}^{2}θ+\text{tan}^{4}θ}$

Show that

\dfrac{\sqrt{2} \; \text{cos} \left( \theta + \dfrac{\pi}{4} \right)}{\text{sin} \left( \theta - \dfrac{\pi}{2} \right)} \equiv \text{tan} \; \theta - 1

Show that

$\text{cot}^{2} θ - \text{tan}^{2} θ \equiv 4 \text{cot} 2 θ \text{cosec} 2 θ$

Show that

$\frac{1-\text{tan}^{2}x}{\text{cos}2x} \equiv \text{sec}^{2} x$

where $x \neq \frac{(2k+1)π}{4}$ and $k$ is an integer.

Show that

$\text{cosec} x \equiv \frac{\frac{1}{2}\text{sec}^{2}\frac{x}{2}}{\text{tan}\frac{x}{2}}$

Show that

$\text{tan} \frac{x}{2} \equiv \frac{1}{\text{cosec}x+\text{cot}x}$

where $x \neq 2 k π$ and $k$ is an integer.

Show that

$- 16 \text{cot} 2 θ \text{cosec}^{3} 2 θ \equiv \text{sec}^{4} θ - \text{cosec}^{4} θ$

Show that

4 \; \text{cos}^2 \left( x - \dfrac{\pi}{6} \right) \equiv 3 - 2 \; \text{sin}^2 \; x + \sqrt{3} \; \text{sin} \; 2x

Show that

\text{tan} \left( \dfrac{2x + \pi}{4} \right) \equiv \text{sec} \; x + \text{tan} \; x

Show that

$\text{sin} 3 θ \equiv 3 \text{sin} θ \text{cos}^{2} θ - \text{sin}^{3} θ$

Hence, or otherwise, show that

$\frac{\text{cos}3θ-\text{cos}θ}{\text{sin}3θ\text{sin}θ} \equiv \frac{4\text{cos}θ}{1-4\text{cos}^{2}θ}$

where $θ \neq k π$ and $k$ is an integer.

Show that

\dfrac{1}{\left( \dfrac{\sqrt{3}}{2} \; \text{cos} \; \theta - \dfrac{1}{2} \; \text{sin} \; \theta \right)^2} + \dfrac{1}{\left( \dfrac{\sqrt{3}}{2} \; \text{sin} \; \theta + \dfrac{1}{2} \; \text{cos} \; \theta \right)^2} \equiv 4 \; \text{cosec}^2 \left( 2\theta + \dfrac{\pi}{3} \right)