Draw a horizontal line (the "x"-axis); mark an origin O. Draw a line from O at an angle formula_37 above the horizontal line and a second line at an angle formula_38 above that; the angle between the second line and the "x"-axis is formula_39.

Place P on the line defined by formula_39 at a unit distance from the origin.

Let PQ be a line perpendicular to line defined by angle formula_37, drawn from point Q on this line to point P. formula_42 OQP is a right angle.

Let QA be a perpendicular from point A on the "x"-axis to Q and PB be a perpendicular from point B on the "x"-axis to P. formula_42 OAQ and OBP are right angles.

Draw QR parallel to the "x"-axis.

Now angle formula_44 (because formula_45, 
making formula_46, and finally formula_44) 

By substituting formula_57 for formula_38 and using Symmetry, we also get:

Another simple "proof" can be given using Euler's formula known from complex analysis:
Euler's formula is:

Although it is more precise to say that Euler's formula entails the trigonometric identities, it follows that for angles formula_37 and formula_38 we have:

Also using the following properties of exponential functions:

Evaluating the product:

Equating real and imaginary parts:

विविध
कोज्या (Cosine)
कोज्या सर्वसमिकाओं की उपपत्ति
ज्या (Sine)