Trig Identities
𝑠𝑖𝑛(𝑥)
tan(x) = 𝑐𝑜𝑠(𝑥)
1
cosec(x) = 𝑠𝑖𝑛(𝑥)
𝑠𝑖𝑛 → 𝑐𝑜𝑠𝑒𝑐
1
sec(x) = 𝑐𝑜𝑠(𝑥)
𝑐𝑜𝑠 → 𝑠𝑒𝑐
1 𝑐𝑜𝑠(𝑥)
cot(x) = 𝑡𝑎𝑛(𝑥)
= 𝑠𝑖𝑛(𝑥) 𝑡𝑎𝑛 → 𝑐𝑜𝑡
Identity Rules
2 2
𝑠𝑖𝑛 (𝑥) + 𝑐𝑜𝑠 (𝑥) = 1
2 2
⇒ 1 − 𝑠𝑖𝑛 (𝑥) = 𝑐𝑜𝑠 (𝑥)
2 2
⇒ 1 − 𝑐𝑜𝑠 (𝑥) = 𝑠𝑖𝑛 (𝑥)
2 2
𝑡𝑎𝑛 (𝑥) + 1 = 𝑠𝑒𝑐 (𝑥)
2 2
⇒ 𝑡𝑎𝑛 (𝑥) = 𝑠𝑒𝑐 (𝑥) − 1
2 2
𝑐𝑜𝑡 (𝑥) + 1 = 𝑐𝑜𝑠𝑒𝑐 (𝑥)
Complementary Angles:
𝑠𝑖𝑛θ = 𝑐𝑜𝑠(90 − θ)
𝑐𝑜𝑠θ = 𝑠𝑖𝑛(90 − θ)
𝑡𝑎𝑛θ = 𝑐𝑜𝑡(90 − θ)
Compound Angles:
𝑠𝑖𝑛(α + β) = 𝑠𝑖𝑛α 𝑐𝑜𝑠β + 𝑐𝑜𝑠α 𝑠𝑖𝑛β
𝑠𝑖𝑛(α − β) = 𝑠𝑖𝑛α 𝑐𝑜𝑠β − 𝑐𝑜𝑠α 𝑠𝑖𝑛β
𝑐𝑜𝑠(α + β) = 𝑐𝑜𝑠α 𝑐𝑜𝑠β − 𝑠𝑖𝑛α 𝑠𝑖𝑛β
𝑐𝑜𝑠(α − β) = 𝑐𝑜𝑠α 𝑐𝑜𝑠β + 𝑠𝑖𝑛α 𝑠𝑖𝑛β
𝑡𝑎𝑛α + 𝑡𝑎𝑛β
𝑡𝑎𝑛(α + β) = 1− 𝑡𝑎𝑛α 𝑡𝑎𝑛β
𝑡𝑎𝑛α − 𝑡𝑎𝑛β
𝑡𝑎𝑛(α − β) = 1+ 𝑡𝑎𝑛α 𝑡𝑎𝑛β
Double Angles:
2𝑡𝑎𝑛𝑥
𝑠𝑖𝑛(2𝑥) = 2 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠 𝑥 = 2
1+𝑡𝑎𝑛 𝑥
2 2
𝑐𝑜𝑠(2𝑥) = 𝑐𝑜𝑠 𝑥 − 𝑠𝑖𝑛 𝑥
2
= 2 𝑐𝑜𝑠 𝑥 − 1
2
= 1 − 2 𝑠𝑖𝑛 𝑥
2𝑡𝑎𝑛𝑥
𝑡𝑎𝑛(2𝑥) = 2
1−𝑡𝑎𝑛 𝑥
2
𝑐𝑜𝑡 𝑥−1
𝑐𝑜𝑡(2𝑥) = 2𝑐𝑜𝑡𝑥
, 𝑑
Constant Rule 𝑑𝑥
c=0
𝑑 𝑛 𝑛−1
Power Rule 𝑑𝑥
𝑥 = 𝑛. 𝑥
𝑑
Sum Rule 𝑑𝑥
f(x) + g(x) = f’(x) + g’(x)
𝑑
Difference Rule 𝑑𝑥
f(x) - g(x) = f’(x) - g’(x)
𝑑
Product Rule 𝑑𝑥
(f(x) . g(x))= f’(x).g(x) + f(x).g’(x)
𝑑 𝑓’(𝑥).𝑔(𝑥) − 𝑓(𝑥).𝑔’(𝑥)
Quotient Rule 𝑑𝑥
(f(x) . g(x))= 2
[𝑔(𝑥)]
𝑑
Chain Rule 𝑑𝑥
(f(g(x)) = f’(g(x).g’(x)
𝑑 𝑥 𝑥
Exponential Rule 𝑑𝑥
𝑏 = 𝑏 ln(b)
𝑑 1
Logarithmic Rule 𝑑𝑥
ln(x) = 𝑥
Log Rule Derivatives
𝑑 1
𝑑𝑥
ln(x) = 𝑥
,x>0
𝑑 𝑔'(𝑥)
𝑑𝑥
ln(g(x)) = 𝑔(𝑥)
𝑑 1
𝑑𝑥
𝑙𝑜𝑔𝑎x = 𝑥 𝑙𝑛 𝑎
, x>0
𝑑 𝑔'(𝑥)
𝑑𝑥
𝑙𝑜𝑔𝑎g(x) = 𝑔(𝑥) 𝑙𝑛 𝑎
Exponential Function Derivatives
𝑑 𝑥 𝑥
𝑑𝑥
(𝑒 ) = 𝑒
𝑑 𝑥 𝑥
𝑑𝑥
(𝑎 ) = 𝑎 ln 𝑎
𝑑 𝑔(𝑥) 𝑔(𝑥)
𝑑𝑥
(𝑒 )=𝑒 g ‘ (x)
𝑑 𝑔(𝑥) 𝑔(𝑥)
𝑑𝑥
(𝑎 ) = 𝑙𝑛(𝑎) 𝑎 g ‘(x)
Trigonometric Derivatives
𝑑
𝑑𝑥
sin(x) = cos(x)
𝑑
𝑑𝑥
cos(x) = -sin(x)
𝑑
𝑑𝑥
tan(x) = sec2(x)
𝑑
𝑑𝑥
cosec(x) = - cosec(x) cot(x)
𝑑
𝑑𝑥
sec(x) = sec(x) tan(x)
𝑑
𝑑𝑥
cot(x) = -cosec2(x)
𝑠𝑖𝑛(𝑥)
tan(x) = 𝑐𝑜𝑠(𝑥)
1
cosec(x) = 𝑠𝑖𝑛(𝑥)
𝑠𝑖𝑛 → 𝑐𝑜𝑠𝑒𝑐
1
sec(x) = 𝑐𝑜𝑠(𝑥)
𝑐𝑜𝑠 → 𝑠𝑒𝑐
1 𝑐𝑜𝑠(𝑥)
cot(x) = 𝑡𝑎𝑛(𝑥)
= 𝑠𝑖𝑛(𝑥) 𝑡𝑎𝑛 → 𝑐𝑜𝑡
Identity Rules
2 2
𝑠𝑖𝑛 (𝑥) + 𝑐𝑜𝑠 (𝑥) = 1
2 2
⇒ 1 − 𝑠𝑖𝑛 (𝑥) = 𝑐𝑜𝑠 (𝑥)
2 2
⇒ 1 − 𝑐𝑜𝑠 (𝑥) = 𝑠𝑖𝑛 (𝑥)
2 2
𝑡𝑎𝑛 (𝑥) + 1 = 𝑠𝑒𝑐 (𝑥)
2 2
⇒ 𝑡𝑎𝑛 (𝑥) = 𝑠𝑒𝑐 (𝑥) − 1
2 2
𝑐𝑜𝑡 (𝑥) + 1 = 𝑐𝑜𝑠𝑒𝑐 (𝑥)
Complementary Angles:
𝑠𝑖𝑛θ = 𝑐𝑜𝑠(90 − θ)
𝑐𝑜𝑠θ = 𝑠𝑖𝑛(90 − θ)
𝑡𝑎𝑛θ = 𝑐𝑜𝑡(90 − θ)
Compound Angles:
𝑠𝑖𝑛(α + β) = 𝑠𝑖𝑛α 𝑐𝑜𝑠β + 𝑐𝑜𝑠α 𝑠𝑖𝑛β
𝑠𝑖𝑛(α − β) = 𝑠𝑖𝑛α 𝑐𝑜𝑠β − 𝑐𝑜𝑠α 𝑠𝑖𝑛β
𝑐𝑜𝑠(α + β) = 𝑐𝑜𝑠α 𝑐𝑜𝑠β − 𝑠𝑖𝑛α 𝑠𝑖𝑛β
𝑐𝑜𝑠(α − β) = 𝑐𝑜𝑠α 𝑐𝑜𝑠β + 𝑠𝑖𝑛α 𝑠𝑖𝑛β
𝑡𝑎𝑛α + 𝑡𝑎𝑛β
𝑡𝑎𝑛(α + β) = 1− 𝑡𝑎𝑛α 𝑡𝑎𝑛β
𝑡𝑎𝑛α − 𝑡𝑎𝑛β
𝑡𝑎𝑛(α − β) = 1+ 𝑡𝑎𝑛α 𝑡𝑎𝑛β
Double Angles:
2𝑡𝑎𝑛𝑥
𝑠𝑖𝑛(2𝑥) = 2 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠 𝑥 = 2
1+𝑡𝑎𝑛 𝑥
2 2
𝑐𝑜𝑠(2𝑥) = 𝑐𝑜𝑠 𝑥 − 𝑠𝑖𝑛 𝑥
2
= 2 𝑐𝑜𝑠 𝑥 − 1
2
= 1 − 2 𝑠𝑖𝑛 𝑥
2𝑡𝑎𝑛𝑥
𝑡𝑎𝑛(2𝑥) = 2
1−𝑡𝑎𝑛 𝑥
2
𝑐𝑜𝑡 𝑥−1
𝑐𝑜𝑡(2𝑥) = 2𝑐𝑜𝑡𝑥
, 𝑑
Constant Rule 𝑑𝑥
c=0
𝑑 𝑛 𝑛−1
Power Rule 𝑑𝑥
𝑥 = 𝑛. 𝑥
𝑑
Sum Rule 𝑑𝑥
f(x) + g(x) = f’(x) + g’(x)
𝑑
Difference Rule 𝑑𝑥
f(x) - g(x) = f’(x) - g’(x)
𝑑
Product Rule 𝑑𝑥
(f(x) . g(x))= f’(x).g(x) + f(x).g’(x)
𝑑 𝑓’(𝑥).𝑔(𝑥) − 𝑓(𝑥).𝑔’(𝑥)
Quotient Rule 𝑑𝑥
(f(x) . g(x))= 2
[𝑔(𝑥)]
𝑑
Chain Rule 𝑑𝑥
(f(g(x)) = f’(g(x).g’(x)
𝑑 𝑥 𝑥
Exponential Rule 𝑑𝑥
𝑏 = 𝑏 ln(b)
𝑑 1
Logarithmic Rule 𝑑𝑥
ln(x) = 𝑥
Log Rule Derivatives
𝑑 1
𝑑𝑥
ln(x) = 𝑥
,x>0
𝑑 𝑔'(𝑥)
𝑑𝑥
ln(g(x)) = 𝑔(𝑥)
𝑑 1
𝑑𝑥
𝑙𝑜𝑔𝑎x = 𝑥 𝑙𝑛 𝑎
, x>0
𝑑 𝑔'(𝑥)
𝑑𝑥
𝑙𝑜𝑔𝑎g(x) = 𝑔(𝑥) 𝑙𝑛 𝑎
Exponential Function Derivatives
𝑑 𝑥 𝑥
𝑑𝑥
(𝑒 ) = 𝑒
𝑑 𝑥 𝑥
𝑑𝑥
(𝑎 ) = 𝑎 ln 𝑎
𝑑 𝑔(𝑥) 𝑔(𝑥)
𝑑𝑥
(𝑒 )=𝑒 g ‘ (x)
𝑑 𝑔(𝑥) 𝑔(𝑥)
𝑑𝑥
(𝑎 ) = 𝑙𝑛(𝑎) 𝑎 g ‘(x)
Trigonometric Derivatives
𝑑
𝑑𝑥
sin(x) = cos(x)
𝑑
𝑑𝑥
cos(x) = -sin(x)
𝑑
𝑑𝑥
tan(x) = sec2(x)
𝑑
𝑑𝑥
cosec(x) = - cosec(x) cot(x)
𝑑
𝑑𝑥
sec(x) = sec(x) tan(x)
𝑑
𝑑𝑥
cot(x) = -cosec2(x)
