Calc 2 Trig Identities

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Last updated 4:49 PM on 10/2/26
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13 Terms

1
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Reciprocal Identities

csc(x) = 1/sin(x)

sec(x) = 1/cos(x)

cot(x) = 1/tan(x)

2
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Quotient Identities

tan(x) = sin(x)/cos(x)

cot(x) = cos(x)/sin(x)


3
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Pythagorean Identities

sin2(x) + cos2(x) = 1

1 + tan2(x) = sec2(x)

1 + cot2(x) = csc2(x)

4
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Even and Odd Identities

EVEN :

cos(-x) = cos(x)

sec(-x) = sec(x)

cot(-x) = cot(x)

ODD :

sin(-x) = - sin(x)

tan(-x) = - tan(x)

csc(-x) = - csc(x)

5
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Cofunction Identities

sin(x) = cos(π/2 - x)

cos(x) = sin(π/2 - x)

tan(x) = cot(π/2 - x)

cot(x) = tan(π/2 - x)

sec(x) = csc(π/2 - x)

csc(x) = sec(π/2 - x)

6
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Sum and Difference Identities

Sum :

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

tan(A + B) = [tan(A) + tan(B)] / [1 - tan(A)tan(B)]

Difference :

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

tan(A - B) = [tan(A) - tan(B)] / [1 + tan(A)tan(B)]

7
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Double Angle Identities

sin(2x) = 2sin(x)cos(x)

cos(2x) = cos²(x) - sin²(x)

cos(2x) = 2cos²(x) - 1

cos(2x) = 1 - 2sin²(x)

tan(2x) = 2tan(x) / [1 - tan²(x)]

8
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Power Reducing Identities

sin²(x) = [1 - cos(2x)] / 2

cos²(x) = [1 + cos(2x)] / 2

tan²(x) = [1 - cos(2x)] / [1 + cos(2x)]

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Product to Sum Identities

sin(A)sin(B) = [cos(A - B) - cos(A + B)] / 2

cos(A)cos(B) = [cos(A - B) + cos(A + B)] / 2

sin(A)cos(B) = [sin(A + B) + sin(A - B)] / 2

cos(A)sin(B) = [sin(A + B) - sin(A - B)] / 2


10
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Trig Derivatives

d/dx [sin(x)] = cos(x)

d/dx [cos(x)] = - sin(x)

d/dx [tan(x)] = sec²(x)

d/dx [cot(x)] = - csc²(x)

d/dx [sec(x)] = sec(x)tan(x)

d/dx [csc(x)] = - csc(x)cot(x)

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Trig Integrals

∫ sin(x) dx = - cos(x) + C

∫ cos(x) dx = sin(x) + C

∫ sec²(x) dx = tan(x) + C

∫ csc²(x) dx = - cot(x) + C

∫ sec(x)tan(x) dx = sec(x) + C

∫ csc(x)cot(x) dx = - csc(x) + C

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Hyperbolic Functions

sinh(x) = (ex - e-x) / 2

cosh(x) = (ex+e-x) / 2

tanh(x) = sinh(x) / cosh(x)

coth(x) = cosh(x) / sinh(x)

sech(x) = 1 / cosh(x)

csch(x) = 1 / sinh(x)

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Hyperbolic Derivatives

d/dx [sinh(x)] = cosh(x)

d/dx [cosh(x)] = sinh(x)

d/dx [tanh(x)] = sech²(x)

d/dx [coth(x)] = - csch²(x)

d/dx [sech(x)] = - sech(x)tanh(x)

d/dx [csch(x)] = - csch(x)coth(x)