Question

Prove that sinA + sin3A + sin5A + sin7A / cosA + cos3A + cos5A + cos7A = tan4A

Answer 1

sinA+sin3A+sin5A+sin7A/cosA+cos3A+cos5A+cos7A
=(sinA+sin7A)+(sin3A+sin5A)/(cosA+cos7A)+(cos3A+cos5A)
=(2sin4Acos3A+2sin4AcosA)/(2cos4Acos3A+2cos4AcosA)
=2sin4A(cos3A+cosA)/2cos4A(cos3A+cosA)
=sin4A/cos4A
=tan4A (Proved)

Answer 2

To prove:

$\frac{\sin A+\sin 3 A+\sin 5 A+\sin 7 A}{\cos A+\cos 3 A+\cos 5 A+\cos 7 A}=\tan 4 A$

Solution:

$\frac{\sin A+\sin 3 A+\sin 5 A+\sin 7 A}{\cos A+\cos 3 A+\cos 5 A+\cos 7 A}$

$=\frac{(\sin A+\sin 7 A)+(\sin 3 A+\sin 5 A)}{(\cos A+\cos 7 A)+(\cos 3 A+\cos 5 A)}$

$=\frac{(2 \sin 4 A \cos 3 A+2 \sin 4 A \cos A)}{(2 \cos 4 A \cos 3 A+2 \cos 4 A \cos A)}$

$=\frac{2 \sin 4 A(\cos 3 A+\cos A)}{2 \cos 4 A(\cos 3 A+\cos A)}$

$=\frac{\sin 4 \mathrm{A}}{\cos 4 \mathrm{A}}$

$=\tan 4 A$

Hence proved.

“Sine, cosine and tangent” are main functions in trigonometry. We are mainly derived this functions using formulas. ‘Trigonometric functions’ have been ‘extended as functions’ of a “real or complex variable”, which are today ‘pervasive in all mathematics’.