Trigonometric identities of any real number theta:For any real number theta (θ) we shall derive the following fundamental identities.
(1) sin²θ + cos² θ = 1
(2) 1 + tan² θ = sec² θ
(3) 1 + cot² θ = csc ² θ
Trigonometric identities of any real number theta:
Proof : 1
From right angle triangle ABC by Pythagoras theorem, we know that
a² + b² = c²
by dividing both side c², we get
a² / c² + b² /c² = c² / c²
(a / c)²+ (b /c)² = (c / c)²
(sin θ)²+ (cos θ)² = 1
we can write
| sin² θ+ cos² θ = 1 |
Proof : 2
From right angle triangle ABC by Pythagoras theorem, we know that
a² + b² = c²
by dividing both side b², we get
a² / b² + b² /b² = c² / b²
(a / c)²+ (b /b)² = (c / b)²
(tan θ)²+ 1 = (sec θ)²
we can write
| tan² θ+ 1 = sec² θ |
Proof : 3
From right angle triangle ABC by Pythagoras theorem, we know that
a² + b² = c²
by dividing both side a², we get
a² / a² + b² /a² = c² / a²
(a / a)²+ (b /a)² = (c / a)²
1+ (cot θ)² = (sin θ)²
we can write
| 1+ cot² θ = cosec² θ |
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