Prove that sin 70-cos 20- - cosec 20- 70- - 2 cos 70-cosec 20- - 0

To prove that, sin 70^∘cos 20^∘ + cosec 20^∘ 70^∘ 2 cos 70^∘ cosec 20^∘ #160; = 0 We know that, #10; #10; cos(90 θ)=sinθ #10; #10; (90 ...

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Standard X

Mathematics

Complementary Trigonometric Ratios

Prove that ...

Question

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

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Solution

To prove that,

$\frac{sin 70^{\circ}}{cos 20^{\circ}} + \frac{c o s e c 20^{\circ}}{sec 70^{\circ}} - 2 cos 70^{\circ} c o s e c 20^{\circ} = 0$

We know that,

$cos (90 - θ) = sin θ$

$sec (90 - θ) = c o s e c θ$

$c o s e c (θ) = \frac{1}{sin θ}$

$\frac{sin 70^{\circ}}{cos (90 - 70)^{\circ}} + \frac{csc 20^{\circ}}{sec (90 - 20)^{\circ}} - 2 \frac{cos 70^{\circ}}{sin 20^{\circ}} = 0$

$\frac{sin 70^{\circ}}{cos (90 - 70)^{\circ}} + \frac{csc 20^{\circ}}{sec (90 - 20)^{\circ}} - 2 \frac{cos 70^{\circ}}{sin (90 - 70^{\circ})} = 0$

$\frac{sin 70^{\circ}}{sin 70^{\circ}} + \frac{csc 20^{\circ}}{csc 20^{\circ}} - 2 \frac{cos 70^{\circ}}{cos 70^{\circ}} = 0$

$1 + 1 - 2 (1) = 0$
$2 - 2 = 0$
$0 = 0$

Hence proved

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Similar questions

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

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

sin70/cos20+cosec20/sec70-2cos70cosec20.Simplify

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 :

\frac{3 cos 55^{\circ}}{7 sin 35^{\circ}} - \frac{4 cos 70^{\circ} . cos e c 20^{\circ}}{7 tan 5^{\circ} . tan 25^{\circ} . tan 45^{\circ} . tan 65^{\circ} . tan 85^{\circ}} = \frac{- 1}{7}

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Prove that sin70∘cos20∘+cosec20∘sec70∘−2cos70∘cosec20∘=0

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