Trigonometric Identities

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.

1. Pythagorean Identities:
   • sin2(x)+cos2(x)=11+tan2(x)=sec2(x)1+cot2(x)=cosec2(x)
   These timeless identities, rooted in the Pythagorean theorem, establish the relationships between the sine, cosine, tangent, secant, and cosecant functions. They find applications in diverse fields, from physics to engineering, where understanding the interplay of these functions is crucial.
2. Reciprocal Identities:
   • cosec(x)=sin(x)1​sec(x)=cos(x)1​cot(x)=tan(x)1​
   These identities unveil the reciprocal nature of trigonometric functions, showcasing how each function’s value relates to its reciprocal counterpart. They are instrumental in simplifying complex trigonometric expressions and solving equations efficiently.
3. Co-function Identities:
   • sin(2π​−x)=cos(x)tan(2π​−x)=cot(x)sec(2π​−x)=cosec(x)
   Co-function identities highlight the symmetry between sine, cosine, tangent, secant, cotangent, and cosecant functions when their arguments complement each other. This symmetry aids in simplifying trigonometric expressions and proofs.
4. 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)​
   These identities, crucial in trigonometric transformations, enable us to express the sine, cosine, and tangent of sums and differences of angles in terms of simpler trigonometric functions.
5. Double Angle Identities:
   • sin(2x)=2sin(x)cos(x)cos(2x)=cos2(x)−sin2(x)tan(2x)=1−tan2(x)2tan(x)​
   Double angle identities provide formulas to express the sine, cosine, and tangent of double angles in relation to single angles, facilitating the simplification of trigonometric expressions involving double angles.

6. 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(2x+y​)cos(2x−y​)
   These identities bridge the gap between products and sums of trigonometric functions, offering powerful tools for simplifying and transforming trigonometric expressions involving products or sums.
7. Half Angle Identities:
8. Common Half angle identity:
9.  sina=2sin(a2)⋅cos(a2)

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

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

6. 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.

6. Power-Reducing Formulas:

• sin² θ = (1 – cos 2θ)/2.
• cos² θ = (1 + cos 2θ)/2.
• tan² θ = (1 – cos 2θ)/(1 + cos 2θ)
• cosec² θ = 2/(1 – cos 2θ)
• sec² θ = 2/(1 + cos 2θ)
• cot² θ = (1 + cos 2θ)/(1 – cos 2θ)

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

6. Inverse Trigonometric Identities:

6. sin−1(x)+cos−1(x)=π/2,​ for x in [−1,1][−1,1]
7. 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.

6. 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