Question

cosA + cos3A +cos5A+ cos7A/ sinA + sin3A + sin 5A+ sin7A =cot4A​

Answer 1

Answer:

cosA+cos3A+cos5A+cos7A

sinA+sin3A+sin5A+sin7A

=tan4A

Solution:

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

cosA+cos3A+cos5A+cos7A

sinA+sin3A+sin5A+sin7A

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

(cosA+cos7A)+(cos3A+cos5A)

(sinA+sin7A)+(sin3A+sin5A)

=\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)}=

(2cos4Acos3A+2cos4AcosA)

(2sin4Acos3A+2sin4AcosA)

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

2cos4A(cos3A+cosA)

2sin4A(cos3A+cosA)

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

cos4A

sin4A

=\tan 4 A=tan4A

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’.