Mathematical Induction involving Compound Angle Formula of Tangent

Trigonometric properties and formulae can be used to perform proofs using mathematical induction. In this example, we use the following compound angle formulae for mathematical induction.

$$ \displaystyle \begin{align} \tan (\alpha + \beta) &= \frac{\tan \alpha + \tan \beta}{1 – \tan \alpha \tan \beta} \\ \tan (\alpha – \beta) &= \frac{\tan \alpha – \tan \beta}{1 + \tan \alpha \tan \beta} \end{align} $$

Part 1

Show that \( \displaystyle 1 + \tan n x \tan (n+1)x = \cot x \big[\tan (n+1)x – \tan n x \big] \) using \( \tan (\alpha – \beta) = \displaystyle \frac{\tan \alpha – \tan \beta}{1 + \tan \alpha \tan \beta} \).

\( \displaystyle \begin{align} \tan \big[nx – (n+1)x \big] &= \displaystyle \frac{\tan n x – \tan (n+1)x}{1 + \tan n x \tan (n+1)x} \\ 1 + \tan n x \tan (n+1)x &= \frac{\tan n x – \tan (n+1)x}{\tan \big[nx – (n+1)x \big]} \\ &= \frac{\tan n x – \tan (n+1)x}{\tan (-x)} \\ &= \frac{\tan n x – \tan (n+1)x}{ -\tan x} \\ &= \frac{\tan (n+1)x – \tan n x}{ \tan x} \\ &= \frac{1}{\tan x} \times \big[ \tan (n+1)x – \tan n x \big] \\ \require{AMSsymbols} \therefore 1 + \tan n x \tan (n+1)x &= \cot x \big[ \tan (n+1)x – \tan n x \big] \end{align} \)

Part 2

Use mathematical induction to prove that, for all integers \( n \ge 1 \),

\( \tan x \tan 2x + \tan 2x \tan 3x + \cdots + \tan n x \tan (n+1)x = -(n+1) + \cot x \tan (n+1)x \).

Step 1

\( \begin{align} \text{For } n &= 1: \\ \text{LHS} &= 1 + \tan x \tan 2x \\ &= \cot x (\tan 2x – \tan x) \color{green}{\cdots \text{Part 1}} \\ &= \text{RHS} \\ \require{AMSsymbols} \therefore \text{The } &\text{statement is true for } n=1 \end{align} \)

Step 2

\( \text{Assume the statement is true for } n =k. \)
\( \text{That is, } \tan x \tan 2x + \tan 2x \tan 3x + \cdots + \tan k x \tan (k+1)x = -(k+1) + \cot x \tan (k+1)x \)

Step 3

\( \text{Assume the statement is true for } n =k+1. \)
\( \text{That is, } \tan x \tan 2x + \tan 2x \tan 3x + \cdots + \tan k x \tan (k+1)x + \tan (k+1)x \tan (k+2) x = -(k+2) + \cot x \tan (k+2)x \)
\( \begin{align} \text{LHS} &= \bbox[#F80]{\tan x \tan 2x + \tan 2x \tan 3x + \cdots + \tan k x \tan (k+1)x } + \tan (k+1)x \tan (k+2) x \\ &= \bbox[#F80]{-(k+1) + \cot x \tan (k+1)x } + \tan (k+1)x \tan (k+2) x &\color{green}{\text{Assumption}} \\ &= -(k+1) + \cot x \tan (k+1)x + \tan (k+1)x \tan (k+2) x + 1 – 1 \\ &= -(k+1) + \cot x \tan (k+1)x + \cot x \big[ \tan (k+2)x – \tan (k+1)x \big] – 1 \\ &= -(k+2) + \bbox[#F80]{\cot x \tan (k+1)x} + \cot x \tan (k+2)x – \bbox[#F80]{\cot x \tan (k+1)x} \\ &= -(k+2) + \cot x \tan (k+2)x \\ &= \text{RHS} \end{align} \)
\( \text{The statement is true for } n=k+1. \)
\( \require{AMSsymbols} \therefore \text{The statement is true for } n=1 \text{and for } n=k+1, \text{then it is true for } n=2,3, \cdots \text{ and for all integers } n \ge 1. \)

 

Absolute Value Algebra Arithmetic Mean Arithmetic Sequence Binomial Expansion Binomial Theorem Chain Rule Circle Geometry Combinations Common Difference Common Ratio Compound Angle Formula Compound Interest Cyclic Quadrilateral Differentiation Discriminant Double-Angle Formula Equation Exponent Exponential Function Factorials Functions Geometric Mean Geometric Sequence Geometric Series Inequality Integration Kinematics Logarithm Logarithmic Functions Mathematical Induction Probability Product Rule Proof Quadratic Quotient Rule Rational Functions Sequence Sketching Graphs Surds Transformation Trigonometric Functions Trigonometric Properties VCE Mathematics Volume




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