Solve equation using proven identity

A question is this type if and only if it asks to solve a trigonometric equation by first proving an identity and then using that result (typically marked as 'hence').

3 questions · Standard +0.1

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CAIE P1 2024 June Q5
6 marks Moderate -0.3
5
  1. Prove the identity \(\frac { \sin ^ { 2 } x - \cos x - 1 } { 1 + \cos x } \equiv - \cos x\).
  2. Hence solve the equation \(\frac { \sin ^ { 2 } x - \cos x - 1 } { 2 + 2 \cos x } = \frac { 1 } { 4 }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
    \includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-07_583_990_306_539} The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 } { x ^ { 2 } } + 4\) for \(x < 0\). The diagram shows the graph of \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\).
  3. On this diagram, sketch the graph of \(y = f ^ { - 1 } ( x )\). Show any relevant mirror line.
  4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  5. Solve the equation \(\mathrm { f } ( x ) = 4.5\).
  6. Explain why the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )\) has no solution.
    \includegraphics[max width=\textwidth, alt={}, center]{cacac880-5b44-4fae-8ed8-88a095db69cd-08_522_1036_296_513} In the diagram, \(A O D\) and \(B C\) are two parallel straight lines. Arc \(A B\) is part of a circle with centre \(O\) and radius 15 cm . Angle \(B O A = \theta\) radians. Arc \(C D\) is part of a circle with centre \(O\) and radius 10 cm . Angle \(C O D = \frac { 1 } { 2 } \pi\) radians.
  7. Show that \(\theta = 0.7297\), correct to 4 decimal places.
  8. Find the perimeter and the area of the shape \(A B C D\). Give your answers correct to 3 significant figures.
CAIE P1 2022 November Q7
7 marks Standard +0.3
7
  1. Prove the identity \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } \equiv \frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 }\).
  2. Hence find the exact solutions of the equation \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } = 2\) for \(0 \leqslant \theta \leqslant \pi\).
CAIE P1 2023 November Q7
9 marks Standard +0.3
7
  1. Verify the identity \(( 2 x - 1 ) \left( 4 x ^ { 2 } + 2 x - 1 \right) \equiv 8 x ^ { 3 } - 4 x + 1\).
  2. Prove the identity \(\frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 } \equiv \frac { 1 } { 1 - 2 \cos ^ { 2 } \theta }\).
  3. Using the results of (a) and (b), solve the equation $$\frac { \tan ^ { 2 } \theta + 1 } { \tan ^ { 2 } \theta - 1 } = 4 \cos \theta$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).