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Maths
- Number
  - Types of number
    
    Natural numbers
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    Real numbers
  - Arithmetic
    
    Order of operations
  - Rounding
    
    Rounding to decimal places
    Rounding to nearest integer
    Rounding to significant figures
  - Representing fractions
    
    Converting percentages to decimals
    Converting decimals to percentages
    Converting fractions to decimals
    Converting decimals to fractions
    Converting fractions to percentages
    Converting percentages to fractions
    Converting recurring decimals to fractions
  - Divisibility
    
    Prime numbers
    Composite numbers
    Prime factor decomposition
    Highest common factor
    Lowest common multiple
  - Fractions
    
    Simplifying fractions
    Multiplying fractions
    Dividing fractions
    Adding fractions
    Subtracting fractions
    Algebraic fractions
  - Percentages
    
    Converting percentages to decimals
    Converting decimals to percentages
    Percentage of a quantity
    Percentage increase
    Percentage decrease
  - Ratio
    
    Ratio meaning
    Simplifying ratios
- Algebra
  - Indices and surds
    
    Laws of indices
    Multiplying indices
    Dividing indices
    Raising an index to an index
    Power of 0
    Power of 1
    Fractional powers
    Negative powers
    Changing the base of a power
    Surds
    Simplifying surds
    Adding surds
    Subtracting surds
    Multiplying surds
    Dividing surds
    Rationalising the denominator
  - Linear functions
    
    Equation of a straight line
    Finding the equation of a straight line
    Modulus of a linear function
  - Inequalities
    
    Inequality signs
    Solving linear inequalities
    Quadratic inequalities
  - Polynomial arithmetic
    
    Expanding brackets
    Expanding single brackets
    Expanding double brackets
    Expanding triple brackets
    Expanding brackets with surds
    Factorising quadratics
    Polynomial division
    Difference of two squares
    The factor theorem
  - Quadratics
    
    Quadratic functions and their graphs
    The discriminant of a quadratic equation
    Completing the square
    Solving quadratic equations by completing the square
    Quadratic formula
    Solving disguised quadratics
  - Sequences
    
    Arithmetic sequences
  - Simultaneous equations
    
    Solving simultaneous equations by elimination
    Solving simultaneous equations by substitution
    Number of intersections between graphs
  - Logarithms
    
    The logarithm function
    Laws of logarithms
    Adding logarithms
    Subtracting logarithms
    Multiplying logarithms by a constant
  - Proportionality
    
    Direct proportion
    Inverse proportion
  - Binomial expansion
    
    Binomial expansion
    Binomial expansion by factorial
- Graphs
  - Coordinates
    
    Cartesian coordinates
  - Graph transformations
    
    Translating graphs
    Stretching graphs
    Parallel line equations
    Perpendicular line equations
  - Real-world graphs
    
    Distance-time graph
  - Exponential graphs
    
    Exponential function
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    Logarithm function gradient
    Logarithm graph equation
- Geometry
  - Coordinate geometry
    
    Distance between two points
    Midpoint of points
    Straight line equation
    Circle-line intersection
    Intersection of circles
  - Circles
    
    Equation of a circle
  - Vectors
    
    Position vectors
  - Triangles
    
    Sine rule
  - Trigonometric identities
    
    Finding all sine angles
    Finding all cosine angles
    Finding all tangent angles
    Calculating tangent
    Sine-cosine square identity
    Sine-cosine offset identity
    Differentiating sine
    Differentiating cosine
    Inverse trigonometry
  - Radians
    
    Radian
    Arc length from radians
    Sector area from radians
    Small angle approximation
  - Arcs and sectors
    
    Arc
    Sector
- Calculus
- Further
- Discrete
  - Graphs
    
    Graph vertex
    Graph edge
    Graph walk
    Graph trail
    Graph path
    Graph cycle
    Graph loop
    Simple graph
    Connected graph
    Complete graph
    Graph tree
    Subgraph
    Eulerian graph
    Hamilton graph
    Planar graph
    Adjacency matrix
    Graph complement
    Graph subdivision
    Bipartite graph
    Isomorphic graph
  - Networks
    
    Network node
    Network arc
    Distance matrix
    Minimum spanning tree
    Prim's algorithm
    Kruskal's algorithm
    Network route inspection
  - Activity networks
  - Network flows
    
    Network flows
  - Game theory
    
    Pay-off matrices
    Play safe strategy
    Dominating strategies
    Optimal mixed strategy
  - Linear programming
    
    Linear Programming
- Unsorted
Physics
Chemistry
- Atomic structure
Computing
- Hardware and software
- Operating systems
  - Operating system role
  - OS resource management
- Types of programming language
- Systems software
- Code translation
- Number systems
  - Natural numbers
- Units of data
- Binary
  - Unsigned binary
- Logic gates
- Boolean algebra
RISC-V assembly
C programming
- Hello world
- Comments
- Variables
- Data types
C# programming
- Hello world
- Comments
- Variables
- Data types
- Operators
- Selection
- Iteration
- Arrays
- Methods
- User input
  - Console.ReadLine
- Classes
Minecraft server
- dylancode's Minecraft server optimisation guide
- server.properties
- bukkit.yml
  - spawn-limits
  - ticks-per
- spigot.yml

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On this page

Overview
- Proof

sinh2x identity

Proof

Section titled “Proof”

We know:
So:
That means we’ve shown that !

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