Table of Trigonometric Identities

Reciprocal identities

Pythagorean Identities

Quotient Identities

Co-Function Identities

Even-Odd Identities

Sum-Difference Formulas

Double Angle Formulas

Power-Reducing/Half Angle Formulas

Sum-to-Product Formulas

Product-to-Sum Formulas

Basics

1. (a + b)(a – b) = a² – b²
   
2. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
   
3. (a ± b)² = a² + b²± 2ab
   
4. (a + b + c + d)² = a² + b² + c² + d² + 2(ab + ac + ad + bc + bd + cd)
   
5. (a ± b)³ = a³ ± b³ ± 3ab(a ± b)
   
6. (a + b)(a² + b² - ab) = a³ + b³
   
7. (a + b + c)(a² + b² + c² -ab – bc – ca) = a³ + b³ + c³ – 3abc =
   
   1/2 (a + b + c)[(a - b)² + (b - c)² + (c - a)²]
   
   
8. when a + b + c = 0, a³ + b³ + c³ = 3abc
   
9. (x + a)(x + b) (x + c) = x³ + (a + b + c) x² + (ab + bc + ac)x + abc
   
10. (x – a)(x – b) (x – c) = x³ – (a + b + c) x² + (ab + bc + ac)x – abc
    
11. a⁴ + a²b² + b⁴ = (a² + ab + b²)( a² – ab + b²)
    
12. a⁴ + b⁴ = (a² – √2ab + b²)( a² + √2ab + b²)
    
13. aⁿ + bⁿ = (a + b) (a ^n-1 – a ^n-2 b +  a ^n-3 b² – a ^n-4 b³ +…….. + b ^n-1)
    (valid only if n is odd)
14. aⁿ – bⁿ = (a – b) (a ^n-1 + a ^n-2 b +  a ^n-3 b² + a ^n-4 b³ +……… + b ^n-1)
    {where n ϵ N)
15. (a ± b)²ⁿ is always positive while -(a ± b)²ⁿ is always negative, for any real values of a and b
    
16. (a – b)²ⁿ = (b – a)²” and (a – b)²ⁿ⁺¹ = – (b – a)²ⁿ⁺¹
    
17. if α and β are the roots of equation ax² + bx + c = 0, roots of cx” + bx + a = 0 are 1/α and 1/β.
    if α and β are the roots of equation ax² + bx + c = 0, roots of ax² – bx + c = 0 are -α and -β.
    
18. • n(n + l)(2n + 1) is always divisible by 6.
      
    • 3²ⁿ leaves remainder = 1 when divided by 8
      
    • n³ + (n + 1 )³ + (n + 2 )³ is always divisible by 9
      
    • 10^{2n + 1} + 1 is always divisible by 11
      
    • n(n²- 1) is always divisible by 6
      
    • n²+ n is always even
      
    • 2³ⁿ-1 is always divisible by 7
      
    • 15^2n-1 +l is always divisible by 16
      
    • n³ + 2n is always divisible by 3
      
    • 3⁴ⁿ – 4 ³ⁿ is always divisible by 17
      
    • n! + 1 is not divisible by any number between 2 and n
    (where n! = n (n – l)(n – 2)(n – 3)…….3.2.1)
    
    for eg 5! = 5.4.3.2.1 = 120 and similarly 10! = 10.9.8…….2.1= 3628800
    
    
19. Product of n consecutive numbers is always divisible by n!.
    
20. If n is a positive integer and p is a prime, then n^p – n is divisible by p.
    
21. |x| = x if x ≥ 0 and |x| = – x if x ≤ 0.
    
22. Minimum value of a².sec²Ɵ + b².cosec²Ɵ is (a + b)²; (0° < Ɵ < 90°)
    
    for eg. minimum value of 49 sec²Ɵ + 64.cosec²Ɵ is (7 + 8)² = 225.
23. among all shapes with the same perimeter a circle has the largest area.
    
24. if one diagonal of a quadrilateral bisects the other, then it also bisects the quadrilateral.
    
25. sum of all the angles of a convex quadrilateral = (n – 2)180°
    
26. number of diagonals in a convex quadrilateral = 0.5n(n – 3)
    
27. let P, Q are the midpoints of the nonparallel sides BC and AD of a trapezium ABCD.Then,
    ΔAPD = ΔCQB.