Prove that- sin 3Acos A-cos 3A sin A-2 2A

L.H.S. sin 3Acos A+cos 3Asin A #10; #10; =sin 3Asin A+cos 3AcosAcos Asin A #10; #10; =cos (3A A)cos Asin #10;A ∴ #10;co ...

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Standard XII

Mathematics

Special Integrals - 1

Prove that: ...

Question

Prove that: $\frac{sin 3 A}{cos A} + \frac{cos 3 A}{s i n A} = 2 cot 2 A$

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Solution

L.H.S.
$\frac{sin 3 A}{cos A} + \frac{cos 3 A}{sin A}$

$= \frac{sin 3 A sin A + cos 3 A c o s A}{cos A sin A}$

$= \frac{cos (3 A - A)}{cos A sin A} ∴ cosAcosB + s i n A s i n B = c o s (A - B)$

$= \frac{2 cos 2 A}{2 cos A sin A}$

$= \frac{2 cos 2 A}{sin 2 A}$

$= 2 cot 2 A$
R.H.S.

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Q. Prove that

\frac{S i n^{3} A + c o s^{3} A}{s i n A + c o s A} + \frac{S i n^{3} A - c o s^{3} A}{s i n A - c o s A} = 2

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Prove:

\frac{{c o s}^{3} A + {s i n}^{3} A}{c o s A + s i n A} + \frac{{c o s}^{3} A - {s i n}^{3} A}{c o s A - s i n A} = 2

\frac{{cos}^{3} A + {sin}^{3} A}{cos A + sin A} + \frac{{cos}^{3} A - {sin}^{3} A}{cos A - sin A}

Q. Prove that

\frac{sinA + s i n 3 A + s i n 5 A + s i n 7 A}{c o s A + cos 3 A + cos 5 A + cos 7 A} = tan 4 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) $\frac{s i n (θ + ϕ) - 2 s i n θ + s i n (θ + ϕ)}{c o s (θ + ϕ) - 2 c o s θ + c o s (θ + ϕ)}$

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Prove that: sin3AcosA+cos3A sinA=2cot2A

L.H.S.sin3AcosA+cos3AsinA =sin3AsinA+cos3AcosAcosAsinA =cos(3A−A)cosAsinA∴cosAcosB+sinAsinB=cos(A−B) =2cos2A2cosAsinA =2cos2Asin2A =2cot2A R.H.S.

Prove that: $\frac{sin 3 A}{cos A} + \frac{cos 3 A}{s i n A} = 2 cot 2 A$

L.H.S.
$\frac{sin 3 A}{cos A} + \frac{cos 3 A}{sin A}$

$= \frac{sin 3 A sin A + cos 3 A c o s A}{cos A sin A}$

$= \frac{cos (3 A - A)}{cos A sin A} ∴ cosAcosB + s i n A s i n B = c o s (A - B)$

$= \frac{2 cos 2 A}{2 cos A sin A}$

$= \frac{2 cos 2 A}{sin 2 A}$

$= 2 cot 2 A$
R.H.S.