Further trigonometric identitiesEdexcel International A Level Maths: Subtopic test

1. 1
   The angle $\theta$ is acute and $\tan\theta=\frac{5}{12}$.

   (a)

   Find the value of $\sec\theta$.

   [1 mark]

   • A$\frac{5}{13}$
   • B$\frac{12}{13}$
   • C$\frac{13}{5}$
   • D$\frac{13}{12}$

   (b)

   Find the value of $\operatorname{cosec}\,\theta$.

   [1 mark]

   • A$\frac{13}{12}$
   • B$\frac{13}{5}$
   • C$\frac{12}{5}$
   • D$\frac{5}{13}$

   (c)

   Find the exact value of $\sec\theta-\tan\theta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Throughout, $x$ is any angle for which the expressions are defined.

   (a)

   Which expression is equivalent to $\sec^2x-1$?

   [1 mark]

   • A$\cos^2x$
   • B$\sin^2x$
   • C$\tan^2x$
   • D$\cot^2x$

   (b)

   Which expression is equivalent to $\cot^2x+1$?

   [1 mark]

   • A$\operatorname{cosec}^2x$
   • B$\sec^2x$
   • C$\tan^2x$
   • D$\sin^2x$

   (c)

   Show that $\dfrac{\sec^2x-1}{\sec^2x}\equiv\sin^2x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Angles are measured in degrees and $0\le\theta<360$.

   (a)

   Given that $\sec^2\theta=4\tan\theta-2$, find the two possible values of $\tan\theta$.

   [3 marks]

   (b)

   Hence solve $\sec^2\theta=4\tan\theta-2$, giving your answers to 1 decimal place.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Angles are measured in degrees and a calculator may be used.

   (a)

   (i) Prove that $\dfrac{\tan^2\theta}{\sec\theta+1}\equiv\sec\theta-1$.
   (ii) Hence solve $\dfrac{\tan^2\theta}{\sec\theta+1}=\dfrac12$ for $0\le\theta<360^{\circ}$, giving your answers to 1 decimal place.

   [6 marks]

   (b)

   Solve $\operatorname{cosec}^2\theta=3+\cot\theta$ for $0<\theta<180^{\circ}$, giving your answers to 1 decimal place.

   [6 marks]

   Total for question 4: 12 marks

End of questions