Prove that- i tan 20o-tan-35o-tan-45o-tan-55o-tan-70o-1 ii sin 48o-sec42o-cos-48o-cosec-42o-2 iii sin-70o-cos-20o-cosec-20o-sec-70o-2-cos-70o-cosec-20o-0 iv cos-80o-sin-10o-cos-59o-cosec-31o-2

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Standard X

Mathematics

Complementary Trigonometric Ratios

Prove that: i...

Question

Prove that:

(i) tan $20^{o} t a n 35^{o} t a n 45^{o} t a n 55^{o} t a n 70^{o} = 1$

(ii) sin $48^{o} s e c 42^{o} + c o s 48^{o} c o s e c 42^{o} = 2$

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

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

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Solution

(1)
LHS = tan20°.tan35°.tan45°.tan55°.tan70°
= tan(90 - 70).tan(90-55).tan45°.tan55°.tan70°
we know,
tan(90-∅) = cot∅ use , this concept here,

= cot70°.cot55°.tan45°.tan55°.tan70°
= { tan70°.cot70°} {tan55°.cot55°}.tan45°
we know,
tan∅.cot∅ = 1

= 1 × 1 × 1 = 1 = RHS
Hence LHS = RHS

(2) sec 42 = cosec (90-42) = cosec 48
cosec 42 = sec (90-42) = sec 48

LHS = sin 48 sec 42 + cos 48 cosec 42
= sin 48 cosec 48 + cos 48 sec 48
= 1 + 1 = 2 ( sin 48 cosec 48 =1 and cos 48 sec 48 =1)
RHS = 2
LHS =RHS
Hence proved

(3)
$fraction numerator sin space 70 over denominator cos space 20 end fraction plus fraction numerator cos begin display style e end style begin display style c end style begin display style space end style begin display style 20 end style over denominator s e c space 70 end fraction minus space 2 space cos space 70 space cos e c space 20 equals 0 fraction numerator sin begin display style space end style begin display style 70 end style over denominator sin begin display style space end style begin display style left parenthesis end style begin display style 90 end style begin display style minus end style begin display style 20 end style begin display style right parenthesis end style end fraction plus cos e c space 20 space cos space 70 space minus space 2 space cos space 70 space cos e c space 20 equals 0 1 space minus space cos e c space 20 space cos space 70 space equals 0 1 minus space fraction numerator cos begin display style space end style begin display style 70 end style over denominator sin begin display style space end style begin display style 20 end style end fraction equals 0 1 space minus space fraction numerator cos begin display style space end style begin display style 70 end style over denominator cos begin display style left parenthesis end style begin display style 90 end style begin display style minus end style begin display style 20 end style begin display style right parenthesis end style end fraction equals 0 1 minus space fraction numerator cos begin display style space end style begin display style 70 end style over denominator cos begin display style space end style begin display style 70 end style end fraction equals 0 1 minus 1 space equals 0 L H S thin space equals R H S H e n c e space p r o v e d$

(4)
$fraction numerator cos space 80 over denominator sin space 10 end fraction plus cos space 59 space cos e c space 31 space equals 2 fraction numerator cos begin display style space end style begin display style 80 end style over denominator cos begin display style space end style begin display style left parenthesis end style begin display style 90 end style begin display style minus end style begin display style 10 end style begin display style right parenthesis end style end fraction plus space fraction numerator begin display style cos space 59 end style over denominator sin begin display style space end style begin display style 31 end style end fraction equals 2 fraction numerator cos begin display style space end style begin display style 80 end style over denominator cos begin display style space end style begin display style 80 end style end fraction plus fraction numerator cos begin display style space end style begin display style 59 end style over denominator cos begin display style space end style begin display style left parenthesis end style begin display style 90 end style begin display style minus end style begin display style 31 end style begin display style right parenthesis end style end fraction equals 2 1 plus space fraction numerator cos begin display style space end style begin display style 59 end style begin display style space end style over denominator cos begin display style space end style begin display style 59 end style end fraction equals 2 1 plus 1 equals 2 L H S space equals R H S H e n c e space V e r i f i e d$

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Prove that:

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

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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

Q. Prove that

\frac{sin 70^{\circ}}{cos 20^{\circ}} + \frac{cosec 20^{\circ}}{sec 70^{\circ}} - 2 cos 70^{\circ} cosec 20^{\circ} = 0

Q. Prove that:
(i) cos 55° + cos 65° + cos 175° = 0

(ii) sin 50° − sin 70° + sin 10° = 0

(iii) cos 80° + cos 40° − cos 20° = 0

(iv) cos 20° + cos 100° + cos 140° = 0

(v)

\sin \frac{5 π}{18} - \cos \frac{4 π}{9} = \sqrt{3} \sin \frac{π}{9}

(vi)

\cos \frac{π}{12} - \sin \frac{π}{12} = \frac{1}{\sqrt{2}}

(vii) sin 80° − cos 70° = cos 50°

(viii) sin 51° + cos 81° = cos 21°

Q. Prove that

sin 48^{\circ} sec 42^{\circ} + cos 48^{\circ} cosec 42^{\circ} = 2

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Prove that: (i) tan 20o tan 35o tan 45o tan 55o tan 70o=1 (ii) sin 48o sec42o+cos 48o cosec 42o=2 (iii) sin 70ocos 20o+cosec 20osec 70o−2 cos 70o cosec 20o=0 (iv) cos 80osin 10o+cos 59o cosec 31o=2