Pythagorean Identities

sin²a + cos ²a = 1       tan²a + 1 = sec²a               cot²a + 1 = cosec²a

sin2a = 2 sina cosa     cos2a = cos²a – sin²a        tan 2a = 2tana / (1 – tan²a)   

cot 2a = (cot²a – 1) / 2 cota

Sum and Difference identities

 For angles x and y, we have the following relationships:

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)        sin(x – y) = sin(x)cos(y) – cos(x)sin(y)         

cos(x + y) = cos(x)cos(y) – sin(x)sin(y)       cos(x – y) = cos(x)cos(y) + sin(x)sin(y)       

tan(x+ y) = tan(x) + tan(y)−tan(x) tan(y)       tan(x-y) = tan(x) − tan(y)+tan(x) tan(y)

Identities

sin²θ + cos²θ = 1       

tan²θ + 1 = sec²θ

cot²θ + 1 = cosec²θ

Basics

If θ is the angle in a right-angled triangle, then 

Sin θ = Perpendicular/Hypotenuse            Cos θ = Base/Hypotenuse

Tan θ = Perpendicular/Base.                     Cot θ = Base/Perpendicular

Sec θ = Hypotenuse/Base                         Cosec θ = Hypotenuse/Perpendicular

Table

Find the value of [cos (90⁰ + A) × sec (360⁰ - A) × tan (180⁰ - A)] / [sec (A - 720⁰) × sin (A + 540⁰) × cot (A - 90⁰)]. 

a) 0     b) 1       c) – 1     d) 2 

Solution:

cos (90⁰ + A) × sec (360⁰ - A) × tan (180⁰ - A) / sec (A - 720⁰) × sin (A + 540⁰) × cot (A - 90⁰) 

             ⇒ (-sin A) × (sec A) × (-tan A) / sec (-(2 × 360⁰ - A)) × sin (3 × 180⁰ + A) × [-cot (90⁰ - A)]

⸫ sec (-θ) = sec θ and cot (-θ) = -cot θ         

⇒ sin A × sec A × tan A / sec A × (-sin A) × (-tan A) 

⇒ sin A × sec A × tan A / sin A × sec A × tan A = 1.

Answer: b

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