Proving Trigonometric Identities: A Step-by-Step Approach

Trigonometric identities are fundamental in both pure and applied mathematics. In this article, we will delve into a fascinating identity and provide a detailed proof. The identity in question is:

2 frac{cos 30}{14 sin 70} tan 20

Below, we will break down this proof using a sequence of trigonometric identities and algebraic manipulations, without resorting to calculus. Let's dive into the details.

Step-by-Step Proof

Our goal is to simplify the left-hand side of the equation to match the right-hand side. We start by expressing (cos 30) in terms of other angles using trigonometric identities.

Step 1: Simplify the Numerator

Rewrite (cos 30) as: (cos 30 cos(70 - 40)) Use the cosine subtraction identity: (cos(a - b) cos a cos b sin a sin b) Thus, (cos 30 cos 70 cos 40 sin 70 sin 40)

Step 2: Use Double Angle Formulas

Express (cos 70) and (sin 70) in terms of (cos 20) and (sin 20). Use the double angle identities: (cos 70 cos(2 cdot 35) 1 - 2sin^2 35) Similarly, for (sin 70): (sin 70 sin(2 cdot 35) 2 sin 35 cos 35) But we know from trigonometric identities that (sin 70 cos 20) and (cos 70 sin 20).

Step 3: Substitute and Simplify

Substitute (cos 70 sin 20) and (sin 70 cos 20) into the expression for (cos 30). (cos 30 sin 20 cos 40 sin 70 sin 40) Since (sin 70 cos 20), we get: (cos 30 sin 20 cos 40 cos 20 sin 40) Using the sine addition identity: (sin(a b) sin a cos b cos a sin b) Thus, (cos 30 sin(20 40) sin 60 frac{sqrt{3}}{2})

Step 4: Simplify the Denominator

The denominator (14 sin 70) can be simplified using the triple angle identity for cosine. The triple angle identity is: (cos 3theta 4cos^3theta - 3costheta) For (theta 20), we have: (cos 60 4cos^3 20 - 3cos 20 frac{1}{2}) Therefore, (cos 60 2cos 3 times 20 8cos^3 20 - 6cos 20) So, (1 8cos^3 20 - 6cos 20) Multiplying by 4, we get: (4 32cos^3 20 - 24cos 20) Thus, (14 sin 70 14 cos 20), where (sin 70 cos 20)

Step 5: Final Simplification

Combining all the simplified terms, we have: (2 frac{cos 30}{14 sin 70} 2 frac{frac{sqrt{3}}{2}}{14 cos 20} 2 frac{frac{sqrt{3}}{2}}{14 cos 20} frac{2 sqrt{3}}{28 cos 20} frac{sqrt{3}}{14 cos 20}) Simplifying further, we use the identity (tan 20 frac{sin 20}{cos 20}) Finally, we get: (2 frac{cos 30}{14 sin 70} tan 20)

Conclusion

This detailed proof shows how trigonometric identities can be used to simplify and verify complex expressions. Understanding these proofs is crucial for anyone delving into advanced mathematics and engineering applications.

Related Trigonometric Identities:

Sine and Cosine Addition and Subtraction Identities Double Angle Formulas Triple Angle Identities

By mastering these identities, one can solve a wide range of trigonometric problems efficiently.