त्रिकोणमितीय सर्वसमिकाएँ

Trigonometric identities are a fundamental aspect of trigonometry, which is the study of the relationships between the angles and sides of triangles. These identities are mathematical equations that involve trigonometric functions, such as sine, cosine, and tangent, and are true for all values of the variables involved.

Trigonometry identities are useful for simplifying expressions, solving equations, and proving mathematical theorems in various fields of science and engineering. Understanding the properties and applications of these identities is essential for students and professionals in fields such as mathematics, physics, and engineering.

Pythagorean Trigonometric Identities

The Pythagorean trigonometric identities in trigonometry are derived from the Pythagoras theorem. The following are the 3 Pythagorean trig identities.

• $cos^{2}A+sin^{2}A=1$
• $1+tan^{2}A=sec^{2}A$
• $cot^{2}A+1=cosec^{2}A$

In $\bigtriangleup ABC$ right angled at B (See Fig. 1) we have

Fig.1 Trigonometric Identities

$AB^{2}+BC^{2}=AC^{2}.......(1)$

Dividing each term of $(1)$ by $AC^{2}$

${\frac {AB^{2}}{AC^{2}}}+{\frac {BC^{2}}{AC^{2}}}={\frac {AC^{2}}{AC^{2}}}$

$\left[{\frac {AB}{AC}}\right]^{2}+\left[{\frac {BC}{AC}}\right]^{2}=\left[{\frac {AC}{AC}}\right]^{2}$

$cos^{2}A+sin^{2}A=1$

This is true for all $A$ such that $0^{\circ }\leq A\leq 90^{\circ }$

Dividing each term of $(1)$ by $AB^{2}$

${\frac {AB^{2}}{AB^{2}}}+{\frac {BC^{2}}{AB^{2}}}={\frac {AC^{2}}{AB^{2}}}$

$\left[{\frac {AB}{AB}}\right]^{2}+\left[{\frac {BC}{AB}}\right]^{2}=\left[{\frac {AC}{AB}}\right]^{2}$

$1+tan^{2}A=sec^{2}A$

This is true for all $A$ such that $0^{\circ }\leq A<90^{\circ }$

Dividing each term of $(1)$ by $BC^{2}$

${\frac {AB^{2}}{BC^{2}}}+{\frac {BC^{2}}{BC^{2}}}={\frac {AC^{2}}{BC^{2}}}$

$\left[{\frac {AB}{BC}}\right]^{2}+\left[{\frac {BC}{BC}}\right]^{2}=\left[{\frac {AC}{BC}}\right]^{2}$

$cot^{2}A+1=cosec^{2}A$

This is true for all $A$ such that $0^{\circ }<A\leq 90^{\circ }$