Trigonometric Proof (Edexcel IAL Maths): Exam Questions

Show that

$\mathrm{cot}θ\quad \equiv\frac{\mathrm{cos}θ}{\mathrm{sin}θ}$

Use the identity

cos$( A + B ) \equiv$cos $A$cos $B-$sin $A$sin $B$

to show that

cos $2A\equiv$cos² $A-$sin² $A$

Show by counter-example that

cos $2 \\theta \text{not} \equiv$cos $θ+$cos $θ$

Prove the identity

$\frac{\mathrm{sin}2θ}{2\mathrm{sin}θ}\equiv\mathrm{cos}θ,θ\neq\mathrm{kπ}$

Show that

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

(i) Use the quotient rule to show that

$\frac{d}{d x} [ \text{cosec} x ] = \frac{- \cos x}{\sin ^{2} x}$

(ii) Hence show that

$\frac{d}{d x} [ \text{cosec} x ] = - \cot x \text{cosec} x$

Show that

$3$ sin $2θ-2$ sin $θ\equiv2$ sin $\\theta ( 3 \cos \\theta - 1 )$

Prove the identity

$2$ cosec $2x$cot $x\equiv$cosec² $x,x\neq\frac{kπ}{2}$

Find the value of

(i) $\arccos ( \cos ( 150^{\circ} ) )$

(ii) $\arcsin ( \sin ( 210^{\circ} ) )$

Explain why the answer to part (a) (ii) is not $210^{\circ}$.

Use the identity

$R$ sin$( \\theta + \ \\alpha ) \equiv R$cos $α$ sin $θ+R$sin $α$ cos $θ$

to show that

$4$ sin( theta + pi / 4 ) === 2 sqrt( 2 ( sin theta + cos theta )

Given the identity

sin² $θ+$cos² $θ\equiv1$

prove the following identities:

(i) sec² $θ\equiv1+$tan² $θ$

(ii) cosec² $θ\equiv1+$cot² $θ$

(i) By using the double angle formula for cosine, prove the identity

cos $4θ\equiv8$ cos⁴ $θ-8$ cos² $θ+1$

(ii) Show by counter-example that

sin $4 \\theta \text{not} \equiv 8$ sin⁴ $θ-8$ sin² $θ+1$

Prove the identity

$\frac{4\mathrm{sin}^{4}θ}{\mathrm{sin}^{2}2θ}\equiv\mathrm{tan}^{2}θθ\neq\mathrm{kπ}$

Show that

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

Use the quotient rule to show that

$\frac{d}{d x} [ \tan x ] = \sec ^{2} x$

Show that

sin $3θ+$sin $θ\equiv4$ sin $θ-4$sin³ $θ$

Prove the identity

$\frac{4\mathrm{cot}x\mathrm{cos}2x}{\mathrm{sin}4x}\equiv\mathrm{cosec}^{2}xx\neq\frac{\mathrm{kπ}}{4}$

Show that

sin( arccos ( - 1 half ) ) = sqrt( 3 sin ( pi / 6 )

Show that

arcsin ( cos (3 pi)/(4) ) = - arcsin ( cos pi / 4 )

Show that

sqrt( 2 sin ( theta - pi / 4 ) === sin theta - cos theta

Given the identity

cos$( A + B ) =$cos $A$ cos $B\quad -$sin $A$ sin $B$

prove the following identities:

(i) cos $2θ\equiv$cos² $θ-$sin² $θ$

(ii) cos $2θ\equiv1-2$sin² $θ$

(iii) cos $2θ\equiv2$cos² $θ-1$

(i) Prove the identity

sin $3θ\equiv3$ sin $θ-4$ sin³ $θ$

(ii) Show by counter-example that

cos $3 \\theta \text{not} \equiv 3$ cos $θ-\quad 4$ cos³ $θ$

Show that

cos $4θ+$cos $\frac{π}{3}\equiv8$ sin⁴ $θ-8$ sin² $θ+\frac{3}{2}$

Prove that

cot² $θ-$tan² $θ\equiv4$ cot $2θ$ cosec $2θ$.

Prove the identity

$\frac{1-\mathrm{tan}^{2}x}{\mathrm{cos}2x}\equiv\mathrm{sec}^{2}xx\neq\frac{2k+1}{4}π$

Prove the identity

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

Show that

tan$\frac{x}{2}\equiv\frac{1}{\mathrm{cosec}x+\mathrm{cot}x}x\neq2\mathrm{kπ}$

Given that $y = \arcsin ( k x )$, where $k$ is a constant, show that x = 1 / k cos ( pi / 2 - y ) .

Hence show that the value of $\mathrm{arcsin}kx+\mathrm{arccos}kx$ is constant and independent of $k$. Find the value of this constant.

Show that

$\frac{10}{4 \cos \\theta + 3 \sin \\theta } \equiv 2 \sec ( \\theta - \ \\alpha )$

where

$\\alpha = \arctan ( \frac{3}{4} )$

Consider the three triangles, all of height 1, as shown below.

By applying the area of a triangle formula $A=\frac{1}{2}\mathrm{ab} \mathrm{sin}C$ to each one, prove that,

sin$( A + B ) \equiv$sin $A$ cos $B+$sin $B$ cos $A$

Briefly explain why this only proves the result for $A$ and $B$ being acute angles.

Prove the identity

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

Prove the identity

$-16$ cot $2θ$ cosec³ $2θ\equiv$sec⁴ $θ-$cosec⁴ $θ$

Show that

(sqrt( 2 cos ( theta + pi / 4 ))/(sin ( theta - pi / 2 )) === tan theta - 1

Show that

sin $3θ\equiv3$ sin $θ$ cos² $θ-$sin³ $θ$

Hence, or otherwise, show that

$\frac{\mathrm{cos}3θ-\mathrm{cos}θ}{\mathrm{sin}3θ\mathrm{sin}θ}\equiv\frac{4\mathrm{cos}θ}{1-4\mathrm{cos}^{2}θ}θ\neq\mathrm{kπ}$

Show that

$4$cos² ( x - pi / 6 ) === 3 - 2sin² $x+\sqrt{3}$ sin $2x$

Show that

tan$( \frac{2 x + \mathrm{\pi } }{4} ) \equiv$sec $x+$tan $x$

On the axes below sketch the graphs of

$y = \arccos ( - x )$ and $y = \left| \arcsin x \right|$

With the help of your sketch, determine the exact solution(s) to the equation

$\arccos ( - x ) = \left| \arcsin x \right|$

What can you say about the solution(s) to the equation

$\left| \arccos x \right| = \arcsin ( - x ) ?$

Justify your answer.

Show that

1 / ( (sqrt( 3)/(2) cos theta - 1 half sin theta ) ^2 + 1 / ( (sqrt( 3)/(2) sin theta + 1 half cos theta ) ^2 === 4 cosec ^2 ( 2 theta + pi / 3 )