Proving That ( tan 45^circ / 2 sqrt{3} - 2sqrt{2} ) is Incorrect

Introduction

The statement that ( tan 45^circ / 2 sqrt{3} - 2sqrt{2} ) is incorrect. This article will reveal the correct proof using trigonometric identities and the half-angle formula, demonstrating that the correct value is ( tan 22.5^circ sqrt{2} - 1 ).

Half-Angle Formula for Tangent

To prove that ( tan 45^circ / 2 sqrt{3} - 2sqrt{2} ) is incorrect, we can use the half-angle formula for tangent:

tan(θ/2) (1 - cos θ) / sin θ

Step 1: Substituting ( θ 45^circ )

Let ( θ 45^circ ). We know that:

cos 45^circ 1 / sqrt{2}

sin 45^circ 1 / sqrt{2}

Step 2: Substitution

Substitute ( θ 45^circ ) into the half-angle formula:

tan 22.5^circ (1 - cos 45^circ) / sin 45^circ (1 - 1 / sqrt{2}) / (1 / sqrt{2})

Step 3: Simplify the Expression

Calculate ( 1 - 1 / sqrt{2} ):

1 - 1 / sqrt{2} sqrt{2} / sqrt{2} - 1 / sqrt{2} (sqrt{2} - 1) / sqrt{2}

Substitute this into the tangent formula:

tan 22.5^circ ((sqrt{2} - 1) / sqrt{2}) / (1 / sqrt{2})

This simplifies to:

tan 22.5^circ sqrt{2} - 1

Step 4: Check if ( sqrt{2} - 1 sqrt{3} - 2sqrt{2} )

To check if ( sqrt{2} - 1 sqrt{3} - 2sqrt{2} ), we can set them equal and simplify:

sqrt{2} - 1 sqrt{3} - 2sqrt{2}

Rearranging gives us:

sqrt{2} 2sqrt{2} sqrt{3} 1

This simplifies to:

3sqrt{2} sqrt{3} 1

Now squaring both sides to eliminate the square roots:

3sqrt{2}^2 (sqrt{3} 1)^2

This results in:

18 3 2sqrt{3} 1

Simplifying gives:

18 4 2sqrt{3}

So we have:

14 2sqrt{3}

Dividing both sides by 2:

7 sqrt{3}

This is not a true statement indicating that ( tan 22.5^circ eq sqrt{3} - 2sqrt{2} ).

Alternative Proof Using Trigonometric Identities

Putting ( theta 45^circ ) in the identity:

cos theta frac{1 - tan ^2 frac{theta}{2}}{1 tan ^2 frac{theta}{2}}

Yields:

begin{aligned} frac{1}{sqrt{2}} cos 45^circ frac{1 - tan ^2 frac{45^circ}{2}}{1 tan ^2 frac{45^circ}{2}} Leftrightarrow sqrt{2} - sqrt{2} tan ^2 frac{45^circ}{2} 1 tan ^2 frac{45^circ}{2} Leftrightarrow sqrt{2} - 1 1 sqrt{2} tan ^2 frac{45^circ}{2} Leftrightarrow tan ^2 frac{45^circ}{2} frac{sqrt{2} - 1}{sqrt{2} 1} cdot frac{sqrt{2} - 1}{sqrt{2} - 1} 3 - 2sqrt{2} Leftrightarrow tan frac{45^circ}{2} sqrt{3 - 2sqrt{2}} sqrt{2} - 1 end{aligned}

This further confirms that the correct value is indeed ( tan 22.5^circ sqrt{2} - 1 ).

Conclusion

The statement that ( tan 45^circ / 2 sqrt{3} - 2sqrt{2} ) is incorrect. The correct value is ( tan 22.5^circ sqrt{2} - 1 ).

Keywords: trigonometric identities, half-angle formula, tangent of 45 degrees