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 identities
An 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)
Identities involving higher-order polynomials Algebraic Identities with Three Variables: These identities involve multiplying binomials or trinomials (expressions with three terms) to get more complex expressions. Examples include:
- 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)
Trigonometric Identities:
Trigonometric identities are equations in trigonometry that hold true for all angles (where the expressions are defined) and involve trigonometric functions like sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These identities are vital tools for simplifying trigonometric expressions, proving other identities, and solving trigonometric equations.
Here are some of the important categories of trigonometric identities:
Pythagorean identities: These identities are based on the Pythagorean theorem and relate the trigonometric functions of an angle. The most common ones are:
- sin²(x) + cos²(x) = 1
- 1 + tan²(x) = sec²(x)
- 1 + cot²(x) = csc²(x)
Co-function Identities: Cofunction identities are a specific type of trigonometric identity that relate the trigonometric functions of complementary angles. Complementary angles are two angles that add up to 90 degrees (π/2 radians).
- sin (π/2 – x) = cos(x)
- cos (π/2 – x) = sin(x)
- tan (π/2 – x) = cot(x)
Even-odd identities: in trigonometry are another set of useful relationships between trigonometric functions. These identities classify the functions based on their behavior when the input angle is negated (multiplied by -1).
- sin (-x) = – sin(x)
- cos (-x) = – cos(x)
- tan (-x) = – tan(x)
Double angle identities: are a specific type of trigonometric identity used to express the sine, cosine, and tangent of double an angle (2θ) in terms of the sine and cosine of the original angle (θ). These identities are incredibly valuable for simplifying trigonometric expressions and solving equations.
- sin (2x) = 2 sin (x) cos (x)
- cos (2x) = cos² (x) – sin² (x)
- tan (2x) = \( \frac{2 tan(x)}{1 – tan²(x)} \)
Things to Remember
1. An identity is an equality which holds good at every value.
2. Every identity is an equality but every equality is not an identity.
3. Identities are used to make calculation easier.
4. There are three standard identities.
5. To prove the equality, one can show LHS = RHS.
Some other useful Identities
- (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)
Proof of (a − b)² = a² − 2ab + b²
This identity helps to expand the square of the difference between the two variables.
L.H.S. = (a – b)²
L.H.S. = (a – b) x (a – b)
L.H.S. = a² – 2ab + b²
L.H.S. = R.H.S.
Proof of (a + b)(a – b) = a² – b²
This identity helps to find the difference between two squares without actually calculating the individual square numbers.
L.H.S. = (a + b)(a – b)
L.H.S. = a² – ab + ba + b²
L.H.S. = a² – b²
L.H.S. = R.H.S.
Hence proved.
Proof of (x + a)(x + b) = x² + x(a + b) + ab
L.H.S. = (x + a)(x + b)
L.H.S. = x² + bx + ax +ab
L.H.S. = x² + x(a + b) + ab
L.H.S. = R.H.S.
Hence proved.
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