Trigonometric Identities

Basic Trigonometric Identities

sin²θ + cos²θ = 1 ; −1 ≤ sin θ ≤ 1 ; −1 ≤ cos θ ≤ 1 ∀ θ ∈ R
sec²θ − tan²θ = 1 ; |sec θ| ≥ 1 ∀ θ ∈ R
cosec²θ − cot²θ = 1 ; |cosec θ| ≥ 1 ∀ θ ∈ R

Important Trigonometric Ratios | Trigonometry

sin n π = 0 ; cos n π = (-1)n ; tan n π = 0 where n ∈ I
\(\sin \frac{(2 n+1) \pi}{2}=(-1)^{n} \quad \& \quad \cos \frac{(2 n+1) \pi}{2}=0\)
sin 15° or \(\sin \frac{\pi}{12}=\frac{\sqrt{3}-1}{2 \sqrt{2}}=\cos 75^{\circ} \text { or } \cos \frac{5 \pi}{12}\);
\(\cos 15^{\circ} \text { or } \cos \frac{\pi}{12}=\frac{\sqrt{3}+1}{2 \sqrt{2}}=\sin 75^{\circ} \text { or } \sin \frac{5 \pi}{12}\)
\(\tan 15^{\circ}=\frac{\sqrt{3}-1}{\sqrt{3}+1}=2-\sqrt{3}=\cot 75^{\circ} ; \tan 75^{\circ}=\frac{\sqrt{3}+1}{\sqrt{3}-1}=2+\sqrt{3}=\cot 15^{\circ}\)
\(\sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} ; \quad \cos \frac{\pi}{8}=\frac{\sqrt{2+\sqrt{2}}}{2} ; \tan \frac{\pi}{8}=\sqrt{2}-1 ; \quad \tan \frac{3 \pi}{8}=\sqrt{2}+1\)
\(\sin \frac{\pi}{10} \text { or } \sin 18^{\circ}=\frac{\sqrt{5}-1}{4} \quad \& \quad \cos 36^{\circ} \quad \text { or } \cos \frac{\pi}{5}=\frac{\sqrt{5}+1}{4}\)

Trigonometric Functions of Allied Angles | Trigonometric Identities

If θ is any angle, then − θ, 90 ± θ, 180 ± θ, 270 ± θ, 360 ± θ etc. are called Allied Angles.

sin (− θ) = − sin θ ; cos (− θ) = cos θ
sin (90°- θ) = cos θ ; cos (90° − θ) = sin θ
sin (90°+ θ) = cos θ ; cos (90°+ θ) = − sin θ
sin (180°− θ) = sin θ ; cos (180°− θ) = − cos θ
sin (180°+ θ) = − sin θ; cos (180°+ θ) = − cos θ
sin (270°− θ) = − cos θ ; cos (270°− θ) = − sin θ
sin (270°+ θ) = − cos θ ; cos (270°+ θ) = sin θ

Trigonometric Functions of Sum or Difference of Two Angles | Trigonometry

sin (A ± B) = sinA cosB ± cosA sinB
cos (A ± B) = cosA cosB ∓ sinA sinB
sin²A − sin²B = cos²B − cos²A = sin (A+B) . sin (A− B)
cos²A − sin²B = cos²B − sin²A = cos (A+B) . cos (A − B)
\(\tan (\mathrm{A} \pm \mathrm{B})=\frac{\tan \mathrm{A} \pm \tan \mathrm{B}}{1 \mp \tan \mathrm{A} \tan \mathrm{B}}[/latexl]
[latex]\cot (\mathrm{A} \pm \mathrm{B})=\frac{\cot \mathrm{A} \cot \mathrm{B} \mp 1}{\cot \mathrm{B} \pm \cot \mathrm{A}}\)

Factorisation of The Sum or Difference of Two Sines or Cosines | Trigonometric Identities

\(\sin C+\sin D=2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}\)
\(\sin C-\sin D=2 \cos \frac{C+D}{2} \sin \frac{C-D}{2}\)
\(\cos C+\cos D=2 \cos \frac{C+D}{2} \cos \frac{C-D}{2}\)
\(\cos C-\cos D=-2 \sin \frac{C+D}{2} \sin \frac{C-D}{2}\)

Transformation of Products Into Sum or Difference of Sines & Cosines | Trigonometric Identities

2 sinA cosB = sin(A+B) + sin(A−B)
2 cosA sinB = sin(A+B) − sin(A−B)
2 cosA cosB = cos(A+B) + cos(A−B)
2 sinA sinB = cos(A−B) − cos(A+B)

Multiple Angles And Half Angles | Trigonometric Identities

sin 2A = 2 sinA cosA; \(\sin \theta=2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\)
cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2 sin²A;
\(\cos \theta=\cos ^{2} \frac{\theta}{2}-\sin ^{2} \frac{\theta}{2}=2 \cos ^{2} \frac{\theta}{2}-1=1-2 \sin ^{2} \frac{\theta}{2}\)
2 cos²A = 1 + cos 2A , 2sin²A = 1 − cos 2A; \(\tan ^{2} A=\frac{1-\cos 2 A}{1+\cos 2 A}\)
\(2 \cos ^{2} \frac{\theta}{2}=1+\cos \theta, 2 \sin ^{2} \frac{\theta}{2}=1-\cos \theta\)
\(\tan 2 \mathrm{A}=\frac{2 \tan \mathrm{A}}{1-\tan ^{2} \mathrm{A}} \quad ; \quad \tan \theta=\frac{2 \tan (\theta / 2)}{1-\tan ^{2}(\theta / 2)}\)
\(\sin 2 A=\frac{2 \tan A}{1+\tan ^{2} A}, \quad \cos 2 A=\frac{1-\tan ^{2} A}{1+\tan ^{2} A}\)
sin 3A = 3 sinA − 4 sin³A
cos 3A = 4 cos³A − 3 cosA
\(\tan 3 A=\frac{3 \tan A-\tan ^{3} A}{1-3 \tan ^{2} A}\)

Three Angles | Trigonometric Identities

(a) \(\tan (\mathrm{A}+\mathrm{B}+\mathrm{C})=\frac{\tan \mathrm{A}+\tan \mathrm{B}+\tan \mathrm{C}-\tan \mathrm{A} \tan \mathrm{B} \tan \mathrm{C}}{1-\tan \mathrm{A} \tan \mathrm{B}-\tan \mathrm{B} \tan \mathrm{C}-\tan \mathrm{C} \tan \mathrm{A}}\)
Note If:
- A+B+C = π then tanA + tanB + tanC = tanA tanB tanC
- A+B+C = \(\frac {\pi}{2}\) then tanA tanB + tanB tanC + tanC tanA = 1
(b) If A + B + C = π then:
- sin2A + sin2B + sin2C = 4 sinA sinB sinC
- sinA + sinB + sinC = 4 cos\(\frac {A}{2}\) cos\(\frac {B}{2}\) cos\(\frac {C}{2}\)

Maximum & Minimum Values of Trigonometric Functions

Min. value of a²tan²θ + b²cot²θ = 2ab where θ ∈ R
Max. and Min. value of acosθ + bsinθ are \(\sqrt{a^{2}+b^{2}} \text { and }-\sqrt{a^{2}+b^{2}}\)
If f(θ) = acos(α + θ) + bcos(β + θ) where a, b, α and β are known quantities then –
\(\sqrt{a^{2}+b^{2}+2 a b \cos (\alpha-\beta)} \leq f(\theta) \leq \sqrt{a^{2}+b^{2}+2 a b \cos (\alpha-\beta)}\)
If α,β ∈ (0, \(\frac {\pi}{2}\)) and α + β = σ (constant) then the maximum values of the expression cosα cosβ, cosα + cosβ, sinα + sinβ and sinα sinβ occurs when α = β = σ/2.
If α,β ∈ (0, \(\frac {\pi}{2}\)) and α + β = σ(constant) then the minimum values of the expression secα + secβ, tanα + tanβ, cosecα + cosecβ occurs when α = β = σ/2.
If A, B, C are the angles of a triangle then maximum value of sinA + sinB + sinC and sinA sinB sinC occurs when A = B = C = 600
In case a quadratic in sinθ or cosθ is given then the maximum or minimum values can be interpreted by making a perfect square.

Sum of sines or cosines of n angles,

\(\begin{array}{l}{\sin \alpha+\sin (\alpha+\beta)+\sin (\alpha+2 \beta)+\ldots \ldots+\sin (\alpha+\overline{n-1} \beta)=\frac{\sin \frac{n \beta}{2}}{\sin \frac{\beta}{2}} \sin \left(\alpha+\frac{n-1}{2} \beta\right)} \\ {\cos \alpha+\cos (\alpha+\beta)+\cos (\alpha+2 \beta)+\ldots \ldots+\cos (\alpha+\overline{n-1} \beta)=\frac{\sin \frac{n \beta}{2}}{\sin \frac{\beta}{\frac{\beta}{2}}} \cos \left(\alpha+\frac{n-1}{2} \beta\right)}\end{array}\)