Standard Identities (मानक पहचान) In Hindi
गणित में कई मानक पहचानें हैं जो विभिन्न गणितीय संक्रियाओं और अवधारणाओं को सरल बनाने में मदद करती हैं।Standard Identities In English
Standard identities, also called algebraic identities, are essentially equations in math that are always true, regardless of what values you plug in for the variables. These identities are incredibly helpful because they allow you to manipulate and simplify algebraic expressions.Standard Algebraic identitiesAn algebraic identity is basically an equation in which L.H.S. equals R.H.S. for all values of the variables.There are two main types of Standard Algebraic identities:Identities involving binomials (expressions with two terms) Algebraic Identities with Two Variables: These identities are formed by multiplying binomials together. Some of the most common examples include:- Square of a sum: (a + b)² = a² + b² + 2ab
- Square of a difference: (a – b)² = a² + b² – 2ab
- Difference of squares: a² – b² = (a + b)(a – b)
- Cube of a sum: (a + b)³ = a³ + b³ + 3ab(a + b)
- Sum of cubes: a³ + b³ = (a + b)(a² – ab + b²)
- Difference of cubes: a³ – b³ = (a + b)(a² – ab + b²)
- sin²(x) + cos²(x) = 1
- 1 + tan²(x) = sec²(x)
- 1 + cot²(x) = csc²(x)
- sin (π/2 – x) = cos(x)
- cos (π/2 – x) = sin(x)
- tan (π/2 – x) = cot(x)
- sin (-x) = – sin(x)
- cos (-x) = – cos(x)
- tan (-x) = – tan(x)
- sin (2x) = 2 sin (x) cos (x)
- cos (2x) = cos² (x) – sin² (x)
- tan (2x) = \( \frac{2 tan(x)}{1 – tan²(x)} \)
- (x + y)² = x² + y² + 2xy = (x-y)² + 4xy
- (x – y)² = x² + y² – 2xy = (x² -y)² – 4xy
- (x + y)² + (x – y)² = 2 (x² + y²)
- (x + y)² – (x – y)² = 4xy
- x² – y² = (x + y)(x – y)
- (a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab (a+b)
- (a – b)³ = a³ + 3a²b + 3ab² – b³ = a³ – b³ – 3ab (a-b)
- (a + b)³ + (a – b)³ = 2(a³ + 3ab²) = 2a (a² + 3b²)
- (a + b)³ – (a – b)³ = 6a²b + 2b³ = 2b (3a² + b²)
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