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

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

 

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Divisibility Proof Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Proof Pythagoras Theorem Quadratic Quadratic Factorise Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




Your email address will not be published. Required fields are marked *