Proving Trigonometric Identities: sin x cdot frac{1}{tan x} cos x

To prove the trigonometric identity sin x cdot frac{1}{tan x} cos x, one can start by recalling the fundamental definition of the tangent function.

Recall of Definitions

The tangent of an angle (x) in a right triangle is defined as the ratio of the opposite side to the adjacent side, which mathematically is written as:

tan x frac{sin x}{cos x}.

Step-by-Step Proof

Using the definition of the tangent function, we can express frac{1}{tan x} as follows:

frac{1}{tan x} frac{cos x}{sin x}.

Now, substitute this expression back into the left-hand side of the identity:

sin x cdot frac{1}{tan x} sin x cdot frac{cos x}{sin x}.

Simplification

Notice that sin x in the numerator and denominator cancels out, assuming sin x neq 0:

sin x cdot frac{cos x}{sin x} cos x.

This simplifies to:

cos x.

Thus, we have shown that:

sin x cdot frac{1}{tan x} cos x.

Therefore, the identity is proven.

Related Proofs and Examples

To further solidify the understanding of these trigonometric identities, here are a few examples:

Example 1: Simplify sin x cdot frac{1}{tan x}. Example 2: Prove the identity cos x cos x.

These examples and the proof above are fundamental to mastering trigonometric identities, which are crucial for solving a wide variety of problems in calculus, physics, and engineering.

Additional Resources

For more information and practice on trigonometric identities, the following resources can be useful:

Math is Fun: Trigonometry Khan Academy: Proving Trigonometric Identities Calculator Soup: Trigonometric Identity Prover

Stay curious!

Written by: Abhinav Rachakonda