Verify the Identity tan(x)=(sin(x))/(cos(x))

Problem

tan(x)=sin(x)/cos(x)

Solution

1. Define the trigonometric functions in terms of a right triangle with an angle x opposite side O adjacent side A and hypotenuse H

2. Express the sine function using these ratios:

sin(x)=O/H

3. Express the cosine function using these ratios:

cos(x)=A/H

4. Substitute these definitions into the right-hand side of the identity:

sin(x)/cos(x)=O/H/A/H

5. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator:

O/H⋅H/A=O/A

6. Identify the resulting ratio as the definition of the tangent function:

tan(x)=O/A

7. Conclude that the left-hand side equals the right-hand side.

Final Answer

tan(x)=sin(x)/cos(x)

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