# Trigonometric Functions of Sum and Difference of Two Angles

• sin (−x) = − sin (x)
• cos (−x) = − cos (x)

Tips:

• sin (A + B) + sin (A − B) = 2 sin A · cos B [A > B]
• cos (A + B) + cos (A − B) = 2 cos A · cos B
• Tan \ C + Tan \ D = \frac{\sin (C + D)}{\cos C \cdot \cos D}
• Tan \ C - Tan \ D = \frac{\sin (C - D)}{\cos C \cdot \cos D}
• \cot \ C + \cot \ D = \frac{\sin (C + D)}{\sin C \cdot \sin D}
• \cot \ C - \cot \ D = \frac{\sin (D - C)}{\sin C \cdot \sin D}
• cos A · cos 2A · cos 4A ...... cos 2n−1·A = \tt \frac{\sin 2^{n}A}{2^{n}\cdot Sin A}
• \tt Sin \frac{\theta}{2}\pm Cos\frac{\theta}{2}=\sqrt{2}\ Sin\left(\frac{\pi}{4}\pm \theta\right)=\sqrt{2}\cdot Cos\left(\theta\mp\frac{\pi}{4}\right)
• The greatest and least values of a sin θ + b cos θ is \sqrt{a^{2}+b^{2}},-\sqrt{a^{2}+b^{2}}
• If A + B + C = π then Sin 2A + Sin 2B + Sin 2C = 4 Sin A · Sin B · Sin C.
• If A + B + C = π then Cos 2A + Cos 2B + Cos 2C = −1 − 4 Cos A Cos B Cos C.
• If A + B + C = π then Tan A + Tan B + Tan C = Tan A · Tan B · Tan C.
• If A + B + C = π then Cot A · Cot B + Cot B · Cot C + Cot C · Cot A = 1
• Tricks:
• \begin{vmatrix}\sin \frac{A}{2}+\cos \frac{A}{2}\end{vmatrix}=\sqrt{1+\sin A} (or) \sin \frac{A}{2}+\cos \frac{A}{2}=\pm\sqrt{1+\sin A} i.e., \tt \begin{cases}+ & if \ 2n\pi-\frac{\pi}{4}\leq A/2 \leq 2n\pi + \frac{3\pi}{4}\\- & Other \ wise\end{cases}
• \begin{vmatrix}\sin \frac{A}{2}-\cos \frac{A}{2}\end{vmatrix}=\sqrt{1-\sin A} (or) \sin \frac{A}{2}-\cos \frac{A}{2}=\pm\sqrt{1-\sin A} [Above condition]
• Tan\frac{A}{2}=\pm\sqrt{\frac{1-\cos A}{1+\cos A}}=\frac{1-\cos A}{\sin A} where A ≠ (2n + 1)π
• \cot\frac{A}{2}=\pm\sqrt{\frac{1+\cos A}{1-\cos A}}=\frac{1+\cos A}{\sin A} where A ≠ 2nπ
• sin2x + cos2x ≥ 2, for every real 'x'
• cos2x + sec2x ≥ 2, for every real 'x'
• Tan2x + cot2x ≥ 2, for every real 'x'

### Part3: View the Topic in this video From 00:40 To 54:25

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1. Formulae for the trigonometric ratios of sum and differences of two angles

a) sin (A + B) = sin A cos B + cos A sin B
b) sin (A − B) = sin A cos B − cos A sin B
c) cos (A + B) = cos A cos B − sin A sin B
d) cos (A − B) = cos A cos B + sin A sin B
e) \tt \tan \left(A+B\right)=\frac{\tan A + \tan B}{1- \tan A \tan B}
f) \tt \tan \left(A-B\right)=\frac{\tan A - \tan B}{1+ \tan A \tan B}
g) \tt \cot \left(A + B \right) = \frac{\cot A \cot B - 1}{\cot A + \cot B}
h) sin (A + B) sin (A − B) = sin2 A − sin2 B = cos2 B − cos2 A
i) cos (A + B) cos (A − B) = cos2 A − sin2 B = cos2 B − sin2 A

2. Trigonometric Ratios of Multiple Angles

a) sin 2A = 2 sin A cos A = \tt \frac{2 \tan A}{1 + \tan^{2}A}0
b) cos 2A = cos2 A − sin2 A = 2 cos2 A − 1 =  \tt 1 - 2 \sin^{2}A=\frac{1 - \tan^{2}A}{1 + \tan^{2}A}0
c) tan 2A = \tt \frac{2 \tan A}{1 - \tan^{2}A}0
d) sin 3A = 3 sin A − 4 sin3 A
e) cos 3A = 4 cos3 A − 3 cos A
f) tan 3 A = \tt \frac{3 \tan A - \tan^{3}A}{1 - 3 \tan^{2}A}
g) sin A = \tt 2 \sin \frac{A}{2} \cos \frac{A}{2} = \frac{2 \tan \frac{A}{2}}{1+ \tan^{2}\frac{A}{2}}
h) cos A = \tt \frac{1-\tan^{2}\frac{A}{2}}{1 + \tan^{2}\frac{A}{2}}
i) 1 − cos A = 2 sin2 \tt \frac{A}{2}
j) 1 + cos A = 2 cos2 \tt \frac{A}{2}
k) \tt \frac{1 - \cos A}{1+ \cos A} = \tan^{2}\frac{A}{2}
l) \tt \sin \left(\frac{A}{2}\right) + \cos \left(\frac{A}{2}\right)=\pm \sqrt{1 + \sin A}
m) \tt \sin \left(\frac{A}{2}\right) - \cos \left(\frac{A}{2}\right)=\pm \sqrt{1 - \sin A}
n) sin 4 θ = 4 sin θ . cos3 θ − 4 cos θ sin3 θ
o) cos 4θ = 8 cos4 θ − 8 cos2 θ + 1
p) tan 4 θ = \tt \frac{4 \tan \theta - 4 \tan^{3} \theta}{1 - 6 \tan^{2} \theta + \tan^{4} \theta}
q) sin 5A = 16 sin5 A − 20 sin3 A + 5 sin A
r) cos 5A = 16 cos5 A − 20 cos3 A + 5 cos A

3. Transformation Formulae

a) 2 sin A cos B = sin (A + B) + sin (A − B)
b) 2 cos A cos B = sin (A + B) − sin (A − B)
c) 2 cos A cos B = cos (A + B) + cos (A − B)
d) 2 sin A sin B = cos (A − B) − cos (A + B)
e) \tt \sin C + \sin D = 2 \sin \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)
f) \tt \sin C − \sin D = 2 \cos \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right)
g) \tt \cos C + \cos D = 2 \cos \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)
h) \tt \cos C − \cos D = -2 \sin \left(\frac{C+D}{2}\right) \sin \left(\frac{C-D}{2}\right) = \sin \left(\frac{C+D}{2}\right) \sin \left(\frac{D-C}{2}\right)