Proving trigonometric identities Calculator

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Example

Created on 2024-06-20Asked by Mia Sanchez (Solvelet student)

Prove the trigonometric identity:

\tan(x) \cdot \cot(x) = 1

Solution

To prove the trigonometric identity

\tan(x) \cdot \cot(x) = 1

: 1. Express

\tan(x)

and

\cot(x)

in terms of sine and cosine:

\tan(x) = \frac{\sin(x)}{\cos(x)}, \quad \cot(x) = \frac{\cos(x)}{\sin(x)}.

2. Multiply

\tan(x)

and

\cot(x)

\tan(x) \cdot \cot(x) = \frac{\sin(x)}{\cos(x)} \cdot \frac{\cos(x)}{\sin(x)}.

3. Simplify the expression:

\tan(x) \cdot \cot(x) = \frac{\sin(x) \cdot \cos(x)}{\cos(x) \cdot \sin(x)} = \frac{\cancel{\sin(x) \cdot \cos(x)}}{\cancel{\cos(x) \cdot \sin(x)}} = 1.

Therefore,

\tan(x) \cdot \cot(x) = 1

, which verifies the identity. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Mateo Scott on Solvelet

1. Prove the identity

\sin(2\theta) = 2\sin(\theta)\cos(\theta)

.2. Verify the identity

\tan^2(\theta) - 1 = \sec^2(\theta) - 1

DefinitionTrigonometric Identities are the expressions and equations which involve trigonometric functions, though are verified only for specific angles and so are true for all the right triangles, theoretical reasonings and declarations at only specific set of angles. Methods including decomposing, simplifying, forcing a common denominator, and using reciprocal, Pythagorean, and co-function identities. For Example: sin2(x)+cos2(x)=1 (Proved by Pythagorean identity.)

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