Trigonometric Identities and Formula: Check Complete List

Trigonometric Identities are fundamental equations in mathematics that relate the six trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent to each other.

Descending Order

Co Prime Numbers

Trigonometric Identities Define

1. Pythagorean Trigonometric Identities

1. sin²(θ) + cos²(θ) = 1 This identity expresses the relationship between the sine and cosine functions. It is one of the fundamental identities and is derived directly from the Pythagorean theorem, where the sum of the squares of the opposite and adjacent sides of a right triangle equals the square of the hypotenuse.
2. 1 + tan²(θ) = sec²(θ) This identity links the tangent and secant functions. It can be derived by dividing the first Pythagorean identity by cos²(θ), leading to the relationship between the secant and tangent functions.
3. cosec²(θ) = 1 + cot²(θ) This identity relates the cosecant and cotangent functions. It is derived by dividing the first Pythagorean identity by sin²(θ), giving the relationship between cosecant and cotangent.

Also Check: Horizontal Line

2. Reciprocal Identities

• sin x = 1/cosec x
• cos x = 1/sec x
• tan x = 1/cot x
• cot x = 1/tan x
• sec x = 1/cos x
• cosec x = 1/sin

3. Complementary and Supplementary Identities

• sin (90°- θ) = cos θ
• cos (90°- θ) = sin θ
• tan (90°- θ) = cot θ
• cosec (90°- θ) = sec θ
• sec (90°- θ) = cosec θ
• cot (90°- θ) = tan θ

• sin (180°- θ) = sinθ
• cos (180°- θ) = -cos θ
• tan (180°- θ) = -tan θ
• cosec (180°- θ) = cosec θ
• sec (180°- θ)= -sec θ
• cot (180°- θ) = -cot θ

• sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta)sin(−θ)=−sin(θ) (Sine is odd)
• cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta)cos(−θ)=cos(θ) (Cosine is even)
• tan⁡(−θ)=−tan⁡(θ)\tan(-\theta) = -\tan(\theta)tan(−θ)=−tan(θ) (Tangent is odd)
• cot⁡(−θ)=−cot⁡(θ)\cot(-\theta) = -\cot(\theta)cot(−θ)=−cot(θ) (Cotangent is odd)
• sec⁡(−θ)=sec⁡(θ)\sec(-\theta) = \sec(\theta)sec(−θ)=sec(θ) (Secant is even)
• csc⁡(−θ)=−csc⁡(θ)\csc(-\theta) = -\csc(\theta)csc(−θ)=−csc(θ) (Cosecant is odd)

5. Sum and Difference Identities

• sinA+sinB=2[sin((A+B)/2)⋅cos((A−B)/2)]
• sinA−sinB=2[cos((A+B)/2)⋅sin((A−B)/2)]
• cosA+cosB=2[cos((A+B)/2)⋅cos((A−B)/2)]
• cosA−cosB=−2[sin((A+B)/2)⋅sin((A−B)/2)]

• cos B = [sin(A + B) + sin(A − B)]/2 cos A⋅
• cos B = [cos(A + B) + cos(A − B)]/2 sin A
• sin B = [cos(A − B) − cos(A + B)]/2

6. Ratio Trigonometric Identities

• Tan θ = Sin θ/Cos θ
• Cot θ = Cos θ/Sin θ

7. Trigonometric Identities of Opposite Angles

• Sin (-θ) = – Sin θ
• Cos (-θ) = Cos θ
• Tan (-θ) = – Tan θ
• Cot (-θ) = – Cot θ
• Sec (-θ) = Sec θ
• Csc (-θ) = -Csc θ

Also Check: Perimeter of Rectangle

8. Trigonometric Identities of Complementary Angles

• Sin (90 – θ) = Cos θ
• Cos (90 – θ) = Sin θ
• Tan (90 – θ) = Cot θ
• Cot ( 90 – θ) = Tan θ
• Sec (90 – θ) = Csc θ
• Csc (90 – θ) = Sec θ

9. Trigonometric Identities of Supplementary Angles

• sin (180°- θ) = sinθ
• cos (180°- θ) = -cos θ
• cosec (180°- θ) = cosec θ
• sec (180°- θ)= -sec θ
• tan (180°- θ) = -tan θ
• cot (180°- θ) = -cot θ

10. Sum and Difference of Angles Trigonometric Identities

• sin(α+β)=sin(α).cos(β)+cos(α).sin(β)
• sin(α–β)=sinα.cosβ–cosα.sinβ
• cos(α+β)=cosα.cosβ–sinα.sinβ
• cos(α–β)=cosα.cosβ+sinα.sinβ
• tan⁡(α+β)=tan⁡α+tan⁡β1–tan⁡α.tan⁡β
• tan⁡(α–β)=tan⁡α–tan⁡β1+tan⁡α.tan⁡β

11. Double Angle Trigonometric Identities

• sin 2θ = 2 sinθ cosθ
• cos 2θ = cos2θ – sin2 θ = 2 cos2θ – 1 = 1 – 2sin2 θ
• tan 2θ = (2tanθ)/(1 – tan2θ)

12. Half Angle Identities

• sin (θ/2) = ±√[(1 – cosθ)/2]
• cos (θ/2) = ±√(1 + cosθ)/2
• tan (θ/2) = ±√[(1 – cosθ)(1 + cosθ)]

13. Product-Sum Trigonometric Identities

• Sin A + Sin B = 2 Sin(A+B)/2 . Cos(A-B)/2
• Cos A + Cos B = 2 Cos(A+B)/2 . Cos(A-B)/2
• Sin A – Sin B = 2 Cos(A+B)/2 . Sin(A-B)/2
• Cos A – Cos B = -2 Sin(A+B)/2 . Sin(A-B)/2