CAT 2025 Lesson : Trigonometry & Coordinate - Trigonometric Formulae

S.No.	Formulae	Expansion
1	sin $^{2}$ θ + cos $^{2}$ θ = 1
2	sin (A + B)	sin A cos B + cos A sin B
3	sin (A – B)	sin A cos B – cos A sin B
4	cos (A + B)	cos A cos B – sin A sin B
5	cos (A – B)	cos A cos B + sin A sin B
6	tan (A + B)	$\dfrac{\mathrm{tan \ A \ + \ tan \ B}}{1 - \mathrm{tan \ A \ tan \ B}}$
7	tan (A – B)	$\dfrac{\mathrm{tan \ A - \ tan \ B}}{1 + \mathrm{tan \ A \ tan \ B}}$
8	sin (A + B) + sin (A – B)	2 sin A cos B
9	sin (A + B) – sin (A – B)	2 cos A sin B
10	cos (A + B) + cos (A – B)	2 cos A cos B
11	cos (A – B) - cos (A + B)	2 sin A sin B
12	sin A + sin B	$2$ sin $\dfrac{\mathrm{A \ + \ B}}{2}$ cos $\dfrac{\mathrm{A \ - \ B}}{2}$
13	sin A – sin B	$2$ cos $\dfrac{\mathrm{A \ + \ B}}{2}$ sin $\dfrac{\mathrm{A \ - \ B}}{2}$
14	cos A + cos B	$2$ cos $\dfrac{\mathrm{A \ + \ B}}{2}$ cos $\dfrac{\mathrm{A \ - \ B}}{2}$
15	cos A – cos B	$-2$ sin $\dfrac{\mathrm{A \ + \ B}}{2}$ sin $\dfrac{\mathrm{A \ - \ B}}{2}$

Additional formulae derived from the above:

S.No.	Formulae	Expansion
1	sin 2A	2 sin A cos A
2	cos 2A	cos $^2$ A - sin $^2$ A = 1 – 2 sin $^2$ A = 2 cos $^2$ A - 1
3	tan 2A	$\dfrac{2 \ \mathrm{tan \ A}}{1 - \tan^{2} \mathrm{A}}$
4	sin 3A	$3 \ \sin \mathrm{A} - 4 \ \sin^{3} \ \mathrm{A}$
5	cos 3A	$4 \ \cos^{3} \mathrm{A} - 3 \ \cos \mathrm{A}$
6	tan 3A	$\dfrac{3 \ \tan \mathrm{A} - \tan^{3} \mathrm{A}}{1 - 3 \tan^{3} \mathrm{A}}$

Find the following
(i) tan

75\degree

(ii) sin

15\degree

+ sin

75\degree

(i)

\tan75\degree = \tan(45\degree + 30\degree) = \dfrac{\tan30\degree + \tan45\degree}{1 - \tan30\degree \tan45\degree} = \dfrac{(\frac{1}{\sqrt{3}}) + 1}{1 - (\frac{1}{\sqrt{3}}) \times 1} = \dfrac{\sqrt{3} + 1}{\sqrt{3} - 1}

\dfrac{\sqrt{3} + 1}{\sqrt{3} - 1} \times \dfrac{\sqrt{3} + 1}{\sqrt{3} + 1} = \dfrac{(\sqrt{3} + 1)^{2}}{(\sqrt{3})^{2} - 1^{2}} = \dfrac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3}

(ii)

\sin 15\degree + \sin 75 \degree = \sin (45\degree - 30\degree) + \sin(45\degree + 30\degree)

\sin 45\degree \ \cos30\degree - \cos45\degree \ \sin 30\degree + \sin 45\degree \ \cos30\degree + \cos45\degree \sin 30\degree

2(\sin 45\degree \cos 30\degree) = 2 \left(\dfrac{1}{\sqrt{2}} \times \dfrac{\sqrt{3}}{2} \right) = \dfrac{\sqrt{3}}{\sqrt{2}} = \sqrt{\dfrac{3}{2}}

Answer: (i)

2 + \sqrt{3}

(ii)

\sqrt{\dfrac{3}{2}}

\cosec 10\degree - \sqrt{3}\sec 10\degree =

(1) 1 (2) 2 (3)

\sqrt{3}

(4) 4

\cosec 10\degree - \sqrt{3}\sec 10\degree = \dfrac{1}{\sin 10\degree} - \dfrac{\sqrt{3}}{\cos 10 \degree} = \dfrac{\cos 10\degree - \sqrt{3} \sin 10\degree}{\sin 10\degree \cos 10\degree}

2 \times \left( \dfrac{1}{2} \times \cos10\degree - \dfrac{\sqrt{3}}{2} \times \sin10\degree \right) \times \dfrac{2}{2 \sin10\degree \ \cos10\degree}

2 \times (\sin 30\degree \ \cos10\degree - \cos 30\degree \ \sin10\degree) \times \dfrac{2}{\sin20\degree}

2 \times \sin (30\degree - 10\degree) \times \dfrac{2}{\sin20\degree} = 4 \times \dfrac{\sin 20\degree}{\sin 20\degree} = 4

Answer: (4) 4

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