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

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Standard X

Mathematics

Relation between Trigonometric Ratios

Prove that :i...

Question

Prove that :

(i) tan 20° tan 35° tan 45° tan 55° tan 70° = 1
(ii) sin 48° sec 42° + cos 48° cosec 42° = 2
(iii) $\frac{\sin 70 °}{\cos 20 °} + \frac{cosec 20 °}{\sec 70 °} - 2 \cos 70 ° cosec 20 ° = 0$
(iv) $\frac{\cos 80 °}{\sin 10 °} + \cos 59 ° cosec 31 ° = 2$

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Solution

We are asked to find the value of

(i) Therefore

Proved

(ii) We will simplify the left hand side

Proved

(iii) We have,

So we will calculate left hand side

Proved

(iv) We have

We will simplify the left hand side

Proved

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Prove that:

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

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

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

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

Q. Prove that:

tan 20^{\circ} tan 35^{\circ} tan 45^{\circ} tan 55^{\circ} tan 70^{\circ} = 1

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Prove that : (i) tan 20° tan 35° tan 45° tan 55° tan 70° = 1 (ii) sin 48° sec 42° + cos 48° cosec 42° = 2 (iii) sin 70°cos 20°+cosec 20°sec 70°-2 cos 70° cosec 20°=0 (iv) cos 80°sin 10°+cos 59° cosec 31°=2

We are asked to find the value of (i) Therefore Proved (ii) We will simplify the left hand side Proved (iii) We have, So we will calculate left hand side Proved (iv) We have We will simplify the left hand side Proved

Prove that :

(i) tan 20° tan 35° tan 45° tan 55° tan 70° = 1
(ii) sin 48° sec 42° + cos 48° cosec 42° = 2
(iii) $\frac{\sin 70 °}{\cos 20 °} + \frac{cosec 20 °}{\sec 70 °} - 2 \cos 70 ° cosec 20 ° = 0$
(iv) $\frac{\cos 80 °}{\sin 10 °} + \cos 59 ° cosec 31 ° = 2$

We are asked to find the value of

(i) Therefore

Proved

(ii) We will simplify the left hand side

Proved

(iii) We have,

So we will calculate left hand side

Proved

(iv) We have

We will simplify the left hand side

Proved