Prove that:
\[\frac{\sin A + 2 \sin 3A + \sin 5A}{\sin 3A + 2 \sin 5A + \sin 7A} = \frac{\sin 3A}{\sin 5A}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Prove that:
\[\frac{\sin \left( \theta + \phi \right) - 2 \sin \theta + \sin \left( \theta - \phi \right)}{\cos \left( \theta + \phi \right) - 2 \cos \theta + \cos \left( \theta - \phi \right)} = \tan \theta\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Prove that:
\[\sin \alpha + \sin \beta + \sin \gamma - \sin (\alpha + \beta + \gamma) = 4 \sin \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\beta + \gamma}{2} \right) \sin \left( \frac{\gamma + \alpha}{2} \right)\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
\[\text{ If } \cos A + \cos B = \frac{1}{2}\text{ and }\sin A + \sin B = \frac{1}{4},\text{ prove that }\tan\left( \frac{A + B}{2} \right) = \frac{1}{2} .\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
\[\text{ If }\sin 2A = \lambda \sin 2B, \text{ prove that }\frac{\tan (A + B)}{\tan (A - B)} = \frac{\lambda + 1}{\lambda - 1}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Prove that:
\[\frac{\cos (A + B + C) + \cos ( - A + B + C) + \cos (A - B + C) + \cos (A + B - C)}{\sin (A + B + C) + \sin ( - A + B + C) + \sin (A - B + C) - \sin (A + B - C)} = \cot C\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Prove that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
\[\text{ If }\frac{\cos (A - B)}{\cos (A + B)} + \frac{\cos (C + D)}{\cos (C - D)} = 0, \text {Prove that }\tan A \tan B \tan C \tan D = - 1\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
If \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
If \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
If (cos α + cos β)2 + (sin α + sin β)2 = \[\lambda \cos^2 \left( \frac{\alpha - \beta}{2} \right)\], write the value of λ.
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
If sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
[3] Trigonometric Functions
Chapter: [3] Trigonometric Functions
Concept: undefined >> undefined
