Trigonometric identities, often hailed as the backbone of trigonometry, are a fascinating realm of mathematical elegance and precision. They form the bedrock upon which complex trigonometric equations and calculations rest, making them indispensable in various mathematical and scientific disciplines.

At first glance, trigonometric identities might seem like a labyrinth of equations and symbols, but delving deeper reveals a world of interconnected patterns and relationships. These identities are essentially equalities involving trigonometric functions, such as sine, cosine, tangent, and their reciprocals.

Let’s embark on a journey to unravel some of the fundamental trigonometric identities and explore their significance in mathematical landscapes.

**Pythagorean Identities:**- sin2(
*x*)+cos2(*x*)=11+tan2(*x*)=sec2(*x*)1+cot2(*x*)=cosec2(*x*)

- sin2(
**Reciprocal Identities:**- cosec(
*x*)=sin(*x*)1sec(*x*)=cos(*x*)1cot(*x*)=tan(*x*)1

- cosec(
**Co-function Identities:**- sin(2
*π*−*x*)=cos(*x*)tan(2*π*−*x*)=cot(*x*)sec(2*π*−*x*)=cosec(*x*)

- sin(2
**Sum and Difference Identities:**- sin(
*x*±*y*)=sin(*x*)cos(*y*)±cos(*x*)sin(*y*)cos(*x*±*y*)=cos(*x*)cos(*y*)∓sin(*x*)sin(*y*)tan(*x*±*y*)=1∓tan(*x*)tan(*y*)tan(*x*)±tan(*y*)

- sin(
**Double Angle Identities:**- sin(2
*x*)=2sin(*x*)cos(*x*)cos(2*x*)=cos2(*x*)−sin2(*x*)tan(2*x*)=1−tan2(*x*)2tan(*x*)

- sin(2

**Product-to-Sum and Sum-to-Product Identities:**- Product-to-Sum: 2sin(
*x*)sin(*y*)=[cos(*x*−*y*)−cos(*x*+*y*)]Sum-to-Product: cos(*x*)+cos(*y*)=2cos(2*x*+*y*)cos(2*x*−*y*)

- Product-to-Sum: 2sin(
**Half Angle Identities:**- Common Half angle identity:
- sina=2sin(a2)⋅cos(a2)

**Half angle Identities in term of t = tan a/2.**

- sina=2t/1+t2
- cosa=1−t2/1+t2
- tana=2t/1−t2

- Half angle identities provide ways to express trigonometric functions of half angles in terms of the original angles, offering insights into angle transformations and simplifications.

**Power-Reducing Formulas:**

- sin
^{2}θ = (1 – cos 2θ)/2. - cos
^{2}θ = (1 + cos 2θ)/2. - tan
^{2}θ = (1 – cos 2θ)/(1 + cos 2θ) - cosec
^{2}θ = 2/(1 – cos 2θ) - sec
^{2}θ = 2/(1 + cos 2θ) - cot
^{2}θ = (1 + cos 2θ)/(1 – cos 2θ)

- These formulas help reduce powers of trigonometric functions, leading to simpler expressions and easier calculations in trigonometric equations.

**Inverse Trigonometric Identities:**

- sin−1(
*x*)+cos−1(*x*)=*π*/2, for*x*in [−1,1][−1,1] - tan−1(x)+cot−1(x)=
*π*/2, for*x*≠0

Inverse trigonometric identities relate the inverses of trigonometric functions, offering insights into the relationships between angles and their trigonometric function values.

**Euler’s Identity:**- eⁱˣ=cos(x)+i⋅sin(x)

Understanding and mastering these trigonometric identities not only enhance problem-solving skills but also pave the way for deeper insights into the mathematical beauty inherent in trigonometry. They serve as powerful tools in tackling complex problems, unleashing the true potential of trigonometric functions in various mathematical contexts.

- sin−1(
*x*)+cos−1(*x*)=*π*/2, for*x*in [−1,1][−1,1] - tan−1(x)+cot−1(x)=
*π*/2, for*x*≠0

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