Evaluate-i 23 cos 430-sin 445-3 sin 260- 245-14 230-ii 4 sin 430-cos 460-23 sin 260-cos 245-12 tan 260-iii sin 50-cos 40-cosec 40- 50-4 cos 50-cosec 40-iv tan 35-tan 40-tan 45-tan 50-tan 55-v cosec65-25-tan55-35-vi tan 7-tan 23-tan 60-tan 67-tan 83-vii 2 sin 68-cos 22-2 15- 5 tan 75-3 tan 45-tan 20-tan 40-tan 50-tan 70- 5viii 3 cos 55- 7 sin 35-4 cos 70-cosec 20- 7 tan 5-tan 25-tan 45-tan 65-tan 85-ix sin 18-cos 72-3 tan 10-tan 30-tan 40-tan 50-tan 80-x cos 58-sin 32-sin 22-cos 68-cos 38-cosec 52-tan 18-tan 35-tan 60-tan 72-tan 55-xi 3 tan 41- 49- 2-sin 35- sec 55-tan 10-tan 20-tan 60-tan 70-tan 80- 2

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Trigonometric Ratios of Compound Angles

Evaluate:i 23...

Question

Evaluate :

(i) $\frac{2}{3} (\cos^{4} 30 - \sin^{4} 45 °) - 3 (\sin^{2} 60 ° - \sec^{2} 45 °) + \frac{1}{4} \cot^{2} 30 °$

(ii) $4 (\sin^{4} 30 ° + \cos^{4} 60 °) - \frac{2}{3} (\sin^{2} 60 ° - \cos^{2} 45 °) + \frac{1}{2} \tan^{2} 60 °$

(iii) $\frac{\sin 50 °}{\cos 40 °} + \frac{cosec 40 °}{\sec 50 °} - 4 \cos 50 ° cosec 40 °$

(iv) tan 35° tan 40° tan 45° tan 50° tan 55°

(v) cosec (65° + θ) − sec (25° − θ) − tan (55° − θ) + cot (35° + θ)

(vi) tan 7° tan 23° tan 60° tan 67° tan 83°

(vii) $\frac{2 \sin 68 °}{\cos 22 °} - \frac{2 \cot 15 °}{5 \tan 75 °} - \frac{3 \tan 45 ° \tan 20 ° \tan 40 ° \tan 50 ° \tan 70 °}{5}$

(viii) $\frac{3 \cos 55 °}{7 \sin 35 °} - \frac{4 (\cos 70 ° cosec 20 °)}{7 (\tan 5 ° \tan 25 ° \tan 45 ° \tan 65 ° \tan 85 °)}$

(ix) $\frac{\sin 18 °}{\cos 72 °} + \sqrt{3} \{\tan 10 ° \tan 30 ° \tan 40 ° \tan 50 ° \tan 80 °\}$

(x) $\frac{\cos 58 °}{\sin 32 °} + \frac{\sin 22 °}{\cos 68 °} - \frac{\cos 38 ° cosec 52 °}{\tan 18 ° \tan 35 ° \tan 60 ° \tan 72 ° \tan 55 °}$

(xi) ${(\frac{3 \tan 41 °}{\cot 49 °})}^{2} - {(\frac{\sin 35 ° \sec 55 °}{\tan 10 ° \tan 20 ° \tan 60 ° \tan 70 ° \tan 80 °})}^{2}$

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Solution

We have to evaluate the following values-

(i) We will use the values of known angles of different trigonometric ratios.

(ii) We will use the values of known angles of different trigonometric ratios.

(iii) We will use the properties of complementary angles.

(iv) We will use the properties of complementary angles.

(v) We will use the properties of complementary angles.

(vi) We will use the properties of complementary angles.

(vii) We will use the properties of complementary angles.

(viii) We will use the properties of complementary angles.

(ix) We will use the properties of complementary angles.

(x) We will use the properties of complementary angles.

(xi)
${(\frac{3 \tan 41 °}{\cot 49 °})}^{2} - {(\frac{\sin 35 ° \sec 55 °}{\tan 10 ° \tan 20 ° \tan 60 ° \tan 70 ° \tan 80 °})}^{2} = {[\frac{3 \tan (90 ° - 49 °)}{\cot 49 °}]}^{2} - {[\frac{\sin 35 ° \sec (90 ° - 35 °)}{\tan 10 ° \tan 20 ° \tan 60 ° \tan (90 ° - 20 °) \tan (90 ° - 10 °)}]}^{2} = {(\frac{3 \cot 49 °}{\cot 49 °})}^{2} - {(\frac{\sin 35 ° cosec 35 °}{\tan 10 ° \tan 20 ° \times \sqrt{3} \times \cot 20 ° \cot 10 °})}^{2} [\tan (90 ° - θ) = \cot θ and \sec (90 ° - θ) = cosec θ]$
$= {(3)}^{2} - {(\frac{1}{\tan 10 ° \cot 10 ° \times \tan 20 ° \cot 20 ° \times \sqrt{3}})}^{2} (cosec θ = \frac{1}{\sin θ}) = 9 - {(\frac{1}{1 \times 1 \times \sqrt{3}})}^{2} (\cot θ = \frac{1}{\tan θ}) = 9 - \frac{1}{3} = \frac{26}{3}$

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Similar questions

(i) $\frac{2}{3} (c o s^{4} 30^{o} - s i n^{4} 45^{o}) - 3 (s i n^{2} 60^{o} - s e c^{2} 45^{o}) + \frac{1}{4} c o t^{2} 30^{o}$

(ii) $4 (s i n^{4} 30^{o} + c o s^{4} 60^{o}) - \frac{2}{3} (s i n^{2} 60^{o} - c o s^{2} 45^{o}) + \frac{1}{2} t a n^{2} 60^{o}$

(iii) $\frac{s i n 50^{o}}{c o s 40^{o}} + \frac{c o s e c 40^{o}}{s e c 50^{o}} - 4 c o s 50^{o} c o s e c 40^{o}$

(iv) $t a n 35^{o} t a n 40^{o} t a n 45^{o} t a n 50^{o} t a n 55^{o}$

(v) $c o s e c (65^{o} + θ) - s e c (25^{o} - θ) - t a n (55^{o} - θ) + c o t (35^{o} + θ)$

(vi) $t a n 7^{o} t a n 23^{o} t a n 60^{o} t a n 67^{o} t a n 83^{o}$

(vii) $\frac{2 s i n 68^{o}}{c o s 22^{o}} - \frac{2 c o t 15^{o}}{5 t a n 75^{o}} - \frac{3 t a n 45^{o} t a n 20^{o} t a n 40^{o} t a n 50^{o} t a n 70^{o}}{5}$
(viii) $\frac{3 c o s 55^{o}}{7 s i n 35^{o}} - \frac{4 (c o s 70^{o} c o s e c 20^{o})}{7 (t a n 5^{o} t a n 25^{o} t a n 45^{o} t a n 65^{o} t a n 85^{o})}$
(ix) $\frac{s i n 18^{o}}{c o s 72^{o}} + \sqrt{3} {t a n 10^{o} t a n 30^{o} t a n 40^{o} t a n 50^{o} t a n 80^{o}}$
(x) $\frac{c o s 58^{o}}{s i n 32^{o}} + \frac{s i n 22^{o}}{c o s 68^{o}} - \frac{c o s 38^{o} c o s e c 52^{o}}{t a n 18^{o} t a n 35^{o} t a n 60^{o} t a n 72^{o} t a n 55^{o}}$

Q. Prove that:

(i) \frac{\sin 70 °}{\cos 20 °} + \frac{cosec 20 °}{\sec 70 °} - 2 \cos 70 ° cosec 20 ° = 0 (ii) \frac{\cos 80 °}{\sin 10 °} + \cos 59 ° cosec 31 ° = 2 (iii) \frac{2 \sin 68 °}{\cos 22 °} - \frac{2 \cot 15 °}{5 \tan 75 °} - \frac{3 \tan 45 ° \tan 20 ° \tan 40 ° \tan 50 ° \tan 70 °}{5} = 1 (iv) \frac{\sin 18 °}{\cos 72 °} + \sqrt{3} (\tan 10 ° \tan 30 ° \tan 40 ° \tan 50 ° \tan 80 °) = 2 [CBSE 2008] (v) \frac{7 \cos 55 °}{3 \sin 35 °} - \frac{4 (\cos 70 ° cosec 20 °)}{3 (\tan 5 ° \tan 25 ° \tan 45 ° \tan 65 ° \tan 85 °)} = 1

Prove that:

(i) $\frac{s i n 70^{\circ}}{c o s 20^{\circ}} + \frac{c o s e c 20^{\circ}}{s e c 70^{\circ}} - 2 c o s 70^{\circ} c o s e c 20^{\circ} = 0$

(ii) $\frac{c o s 80^{\circ}}{s i n 10^{\circ}} + c o s 59^{\circ} c o s e c 31^{\circ} = 2$

(iii) $\frac{2 s i n 68^{\circ}}{c o s 22^{\circ}} - \frac{2 c o t 15^{\circ}}{5 t a n 75^{\circ}} - \frac{3 t a n 45^{\circ} t a n 20^{\circ} t a n 40^{\circ} t a n 50^{\circ} t a n 70^{\circ}}{5} = 1$

(iv) $\frac{s i n 18^{\circ}}{c o s 72^{\circ}} + \sqrt{3} (t a n 10^{\circ} t a n 30^{\circ} t a n 40^{\circ} t a n 50^{\circ} t a n 80^{\circ}) = 2$

(v) $\frac{7 c o s 55^{\circ}}{3 s i n 35^{\circ}} - \frac{4 (c o s 70^{\circ} c o s e c 20^{\circ})}{3 (t a n 5^{\circ} t a n 25^{\circ} t a n 45^{\circ} t a n 65^{\circ} t a n 85^{\circ})} = 1$

tan 10^{\circ} tan 20^{\circ} tan 30^{\circ} tan 40^{\circ} tan 50^{\circ} tan 60^{\circ} tan 70^{\circ} tan 80^{\circ}

is equal to

Q. Evaluate :
(vii)

\frac{2 sin 68^{\circ}}{cos 22^{\circ}} - \frac{2 cot 15^{\circ}}{5 tan 75^{\circ}} - \frac{3 tan 45^{\circ} tan 20^{\circ} tan 40^{\circ} tan 50^{\circ} tan 70^{\circ}}{5}

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