Show that- cos 2A -2cos 4A -cos 6A cos A-2 cos 3A -cos 5A -cos A -sin A -tan 3A

cos2A+2cos4A+cos6A/cosA+2cos3A+cos5A = 2cos (2A+6A2) cos(2A 6A2)+2cos4A2cos (A+5A2) cos(A 5A2)+2cos3A = 2cos4A cos2A+2cos4A2cos3A cos2A+2cos3A ...

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Byju's Answer

Standard XII

Mathematics

Trigonometric Ratios of Compound Angles

Show that: c...

Question

Show that:
$\frac{cos 2 A + 2 cos 4 A + cos 6 A}{cos A + 2 cos 3 A + cos 5 A} = cos A - sin A . tan 3 A$

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Solution

$\frac{c o s 2 A + 2 c o s 4 A + c o s 6 A}{c o s A + 2 c o s 3 A + c o s 5 A}$
$= \frac{2 c o s (\frac{2 A + 6 A}{2}) c o s (\frac{2 A - 6 A}{2}) + 2 c o s 4 A}{2 c o s (\frac{A + 5 A}{2}) c o s (\frac{A - 5 A}{2}) + 2 c o s 3 A}$
$= \frac{2 c o s 4 A c o s 2 A + 2 c o s 4 A}{2 c o s 3 A c o s 2 A + 2 c o s 3 A}$
$= \frac{2 c o s 4 A (c o s 2 A + 1)}{2 c o s 3 A (c o s 2 A + 1)}$
$= c o s 4 A s e c 3 A$
$= c o s A - s i n A t a n 3 A$

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Similar questions

Q. Show that

\frac{cos 2 A + 2 cos 4 A + cos 6 A}{cos A + 2 cos 3 A + cos 5 A} = cos A - sin A \cdot tan 3 A

Q. Prove that

\frac{cos 3 A + 2 cos 5 A + cos 7 A}{cos A + 2 cos 3 A + cos 5 A} = \frac{cos 5 A}{cos 3 A} = cos 2 A - sin 2 A tan 3 A

8. Prove that :
(i) $\frac{s i n A + s i n 3 A + s i n 5 A}{c o s A + c o s 3 A + c o s 5 A} = t a n 3 A$
(ii) $(i i) \frac{c o s 3 A + 2 c o s 5 A + c o s 7 A}{c o s A + 2 c o s 3 A + c o s 5 A} = \frac{c o s 5 A}{c o s 3 A}$
(iii) $\frac{c o s 4 A + c o s 3 A + c o s 2 A}{c o s 4 A + s i n 3 A + s i n 2 A} = c o t 3 A$
(iv) $\frac{s i n 3 A + s i n 5 A + s i n 7 A + s i n 9 A}{c o s 3 A + c o s 5 A + c o s 7 A + c o s 9 A} = t a n 6 A$
(v) $\frac{s i n 5 A - s i n 7 A + s i n 8 A - s i n 4 A}{c o s 4 A + c o s 7 A + c o s 7 A - c o s 5 A - c o s 8 A} = c o t 6 A$
(vi) $\frac{s i n 5 A + c o s 2 A - s i n 6 A c o s A}{s i n A s i n 2 A - c o s 2 A c o s 3 A} = t a n A$
(vii) $\frac{s i n 11 A + s i n A + s i n 7 A + s i n 3 A}{c o s 11 A s i n A + c o s 7 A s i n 3 A} = t a n 8 A$
(viii) $\frac{s i n 3 A c o s 4 A - s i n A c o s 2 A}{s i n 4 A s i n A + c o s 6 A c o s A} = t a n 2 A$
(ix) $\frac{s i n A s i n 2 A + s i n 3 A s i n 6 A}{s i n A c o s 2 A + c o s 3 A c o s 6 A} = t a n 5 A$
(x) $\frac{s i n A + 2 s i n 3 A + s i n 5 A}{s i n 3 A + 2 s i n 5 A + s i n 7 A} = \frac{s i n 3 A}{s i n 5 A}$
(xi) $\frac{s i n (θ + ϕ) - 2 s i n θ + s i n (θ + ϕ)}{c o s (θ + ϕ) - 2 c o s θ + c o s (θ + ϕ)}$

\frac{sin A + sin 2 A + sin 4 A + sin 5 A}{cos A + cos 2 A + cos 4 A + cos 5 A} =

Q. Show that:

\frac{s i n A + s i n 5 A + s i n 9 A}{c o s A + c o s 5 A + c o s 9 A} = t a n 5 A

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Show that:cos2A+2cos4A+cos6AcosA+2cos3A+cos5A=cosA−sinA.tan3A

cos2A+2cos4A+cos6AcosA+2cos3A+cos5A=2cos(2A+6A2)cos(2A−6A2)+2cos4A2cos(A+5A2)cos(A−5A2)+2cos3A=2cos4Acos2A+2cos4A2cos3Acos2A+2cos3A=2cos4A(cos2A+1)2cos3A(cos2A+1)=cos4Asec3A=cosA−sinAtan3A

Show that:
$\frac{cos 2 A + 2 cos 4 A + cos 6 A}{cos A + 2 cos 3 A + cos 5 A} = cos A - sin A . tan 3 A$