Prove that sin A + sin B sin A - sin B = tan A + B 2 cot A - B 2 .

LHS = sin A + sin B sin A - sin B Now using trigonometric identity, sin θ + sin ϕ = 2 sin θ + ϕ 2 cos θ - ϕ 2 & sin θ - sin ϕ = 2 cos θ + ϕ 2 sin θ - ϕ 2 = 2 sin A + B 2 cos A - B 2 2 cos A + B 2 sin A - B 2 = tan A + B 2 cot A - B 2 = RHS Hence, proved.

Prove that sin A + sin B sin A - sin B = tan A + B 2 cot A - B 2 .

LHS = sin A + sin B sin A - sin B Now using trigonometric identity, sin θ + sin ϕ = 2 sin θ + ϕ 2 cos θ - ϕ 2 & sin θ - sin ϕ = 2 cos θ + ϕ 2 sin θ - ϕ 2 = 2 sin A + B 2 cos A - B 2 2 cos A + B 2 sin A - B 2 = tan A + B 2 cot A - B 2 = RHS Hence, proved.

Prove that cos A + cos B cos B - cos A = cot A + B 2 cot A - B 2 .

LHS = cos A + cos B cos B - cos A Apply the identity, cos θ + cos ϕ = 2 cos θ + ϕ 2 cos θ - ϕ 2 & cos θ - cos ϕ = 2 sin θ + ϕ 2 sin ϕ - θ 2 = 2 cos A + B 2 cos A - B 2 2 sin A + B 2 sin A - B 2 = cot A + B 2 cot A - B 2 = RHS Hence proved.

Prove that sin A + sin B cos A + cos B = tan A + B 2 .

sin A + sin B cos A + cos B Apply the identity, sin C + sin D = 2 sin C + D 2 cos C - D 2 & cos C + cos D = 2 cos C + D 2 cos C - D 2 = 2 sin A + B 2 cos A - B 2 2 cos A + B 2 cos A - B 2 = sin A + B 2 cos A + B 2 = tan A + B 2 Hence, proved.

Prove that : sin A - sin B cos B - cos A = cot A + B 2 .

LHS = sin A - sin B cos B - cos A By using trigonometric identity, sin C - sin D = 2 cos C + D 2 sin C - D 2 and cos C - cos D = 2 sin C + D 2 sin D - C 2 = 2 cos A + B 2 sin A - B 2 2 sin A + B 2 sin A - B 2 = cos A + B 2 sin A + B 2 = cot A + B 2 = RHS Hence, proved.

E M B I B E

Plane Trigonometry Part 1>Chapter 7 - Trigonometrical Ratios of The Sum and Difference of Two Angles>EXAMPLES XIV>Q 20

MEDIUM

JEE Main

IMPORTANT

Earn 100

Prove that $\frac{\sin A + \sin B}{\sin A - \sin B} = \tan \frac{A + B}{2} \cot \frac{A - B}{2}$ .

Important Questions on Trigonometrical Ratios of The Sum and Difference of Two Angles

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 7 - Trigonometrical Ratios of The Sum and Difference of Two Angles>EXAMPLES XIV>Q 21

Prove that

\frac{\cos A + \cos B}{\cos B - \cos A} = \cot \frac{A + B}{2} \cot \frac{A - B}{2}

.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 7 - Trigonometrical Ratios of The Sum and Difference of Two Angles>EXAMPLES XIV>Q 22

Prove that

\frac{\sin A + \sin B}{\cos A + \cos B} = \tan \frac{A + B}{2}

.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 7 - Trigonometrical Ratios of The Sum and Difference of Two Angles>EXAMPLES XIV>Q 23

Prove that :

\frac{\sin A - \sin B}{\cos B - \cos A} = \cot \frac{A + B}{2}

.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 7 - Trigonometrical Ratios of The Sum and Difference of Two Angles>EXAMPLES XIV>Q 24

Prove that $\frac{\cos (A + B + C) + \cos (- A + B + C) + \cos (A - B + C) + \cos (A + B - C)}{\sin (A + B + C) + \sin (- A + B + C) - \sin (A - B + C) + \sin (A + B - C)} = \cot B$ .