Identity 1:

Proof 1:

Refer to the triangle diagram above. Note that formula_22 by Pythagorean theorem.

The following two results follow from this and the ratio identities. To obtain the first, divide both sides of formula_24 by formula_25; for the second, divide by formula_26.

Similarly

Proof 2:

Differentiating the left-hand side of the identity yields:

Integrating this shows that the original identity is equal to a constant, and this constant can be found by plugging in any arbitrary value of x.

Identity 2:

The following accounts for all three reciprocal functions.

Proof 1:

Refer to the triangle diagram above. Note that formula_22 by Pythagorean theorem.

Substituting with appropriate functions -

Rearranging gives:

पाइथागोरीय सर्वसमिकाएँ
Prosthaphaeresis identities
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