Proving the Trigonometric Identity: sin x tan x – sin x/tan x 1/cos x

This article delves into the proof of a fundamental trigonometric identity, specifically:

sin x tan x – sin x/tan x 1/cos x

Breakdown and Proof

To prove this identity, we will use several methods to simplify and manipulate the expressions. Let's start with the basic transformation of the tangent function using sine and cosine.

Method 1:

Start with the original expression:
sin x tan x – sin x/tan x Replace tan x with its equivalent sin x/cos x and simplify:
sin x (sin x/cos x) – sin x/(sin x/cos x) This can be rewritten as:
sin x sin x/cos x – sin x cos x/sin x Further simplification gives:
(sin^2 x)/cos x – cos x Combine the terms over a common denominator:
(sin^2 x – cos^2 x) / cos x Utilize the Pythagorean identity, sin^2 x cos^2 x 1, to rewrite the numerator:
(1 – 2cos^2 x 1) / cos x Finally, we achieve the desired result:
1/cos x

Method 2:

Start with the original expression:
sin x tan x – sin x/tan x Express tan x as sin x/cos x:
sin x (sin x/cos x) – sin x/(sin x/cos x) Combine and simplify:
sin x (sin x/cos x) – sin x cos x/sin x This can be simplified to:
(sin^2 x)/cos x – cos x Combine the terms using a common denominator:
(sin^2 x – cos^2 x) / cos x Use the Pythagorean identity to simplify:
(1) / cos x Prove the identity:
1/cos x

In conclusion, we have proven that sin x tan x – sin x/tan x 1/cos x by utilizing basic trigonometric identities and algebraic manipulations. This method not only verifies the identity but also reinforces the understanding of trigonometric functions and their relationships.

Additional Proofs and Verification

To further verify this identity, consider the following steps:

Method 3:

Start with the left-hand side (LHS):
sin x tan x – sin x/tan x Replace tan x with sin x/cos x and simplify:
sin x (sin x/cos x) – sin x/(sin x/cos x) Combine and simplify the terms:
(sin^2 x)/cos x – sin x cos x/sin x This can be written as:
(sin^2 x – cos^2 x) / cos x Utilize the Pythagorean identity again:
(1) / cos x This matches the right-hand side (RHS):
1/cos x

Conclusion

The proofs presented above demonstrate the validity and elegance of the trigonometric identity sin x tan x – sin x/tan x 1/cos x. Understanding and verifying such identities is crucial in advanced mathematics, as they underpin many complex problem-solving techniques in calculus, physics, and engineering. Practice in manipulating these expressions will enhance your problem-solving skills and deepen your understanding of trigonometry.