Without using trigonometric tables, prove that:

(i) $s i n 53^{\circ} c o s 37^{\circ} + c o s 53^{\circ} s i n 37^{\circ} = 1$

(ii) $c o s 54^{\circ} c o s 36^{\circ} - s i n 54^{\circ} s i n 36^{\circ} = 0$

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

(iv) $s i n 35^{\circ} s i n 55^{\circ} - c o s 35^{\circ} c o s 55^{\circ} = 0$

(v) $(s i n 72^{\circ} + c o s 18^{\circ}) (s i n 72^{\circ} - c o s 18^{\circ}) = 0$

(vi) $t a n 48^{\circ} t a n 23^{\circ} t a n 42^{\circ} t a n 67^{\circ} = 1$

Open in App

Solution

(i) $s i n 53^{\circ} c o s 37^{\circ} + c o s 53^{\circ} s i n 37^{\circ} = 1 L H S = s i n 53^{o} c o s 37^{o} + c o s 53^{o} s i n 37^{o} = s i n (90^{o} - 37^{o}) c o s 37^{o} + c o s (90^{o} - 37^{o}) s i n 37^{o} = c o s 37 \circ c o s 37 \circ + s i n 37 \circ s i n 37 \circ = c o s^{2} 37^{o} + s i n^{2} 37^{o} = 1 = R H S$

(ii) $c o s 54^{\circ} c o s 36^{\circ} - s i n 54^{\circ} s i n 36^{\circ} = 0 L H S = c o s 54^{o} c o s 36^{o} - s i n 54^{o} s i n 36^{o} = c o s (90^{o} - 36^{o}) c o s 36^{o} - s i n (90^{o} - 36^{o}) s i n 36^{o} = s i n 36^{o} c o s 36^{o} - c o s 36^{o} s i n 36^{o} = 0 = R H S$

(iii) $s e c 70^{\circ} s i n 20^{\circ} + c o s 20^{\circ} c o s e c 70^{\circ} = 2 L H S = s e c 70^{o} s i n 20^{o} + c o s 20^{o} c o s e c 70^{o} = s e c (90^{o} - 20^{o}) s i n 20^{o} + c o s 20^{o} c o s e c (90^{o} - 20^{o}) = c o s e c 20 v \times \frac{1}{c o s e c 20^{o}} + \frac{1}{s e c 20^{o}} \times s e c 20^{o} = 1 + 1 = 2 = R H S$

(iv) $s i n 35^{\circ} s i n 55^{\circ} - c o s 35^{\circ} c o s 55^{\circ} = 0 L H S = s i n 35^{o} s i n 55^{o} - c o s 35^{o} c o s 55^{o} = s i n 35^{o} c o s (90^{o} - 55^{o}) - c o s 35^{o} s i n (90^{o} - 55^{o}) = s i n 35^{o} c o s 35^{o} - c o s 35^{o} s i n 35 v = 0 = R H S$

(v) $(s i n 72^{\circ} + c o s 18^{\circ}) (s i n 72^{\circ} - c o s 18^{\circ}) = 0 L H S = (s i n 72^{o} + c o s 18^{o}) (s i n 72^{o} - c o s 18^{o}) = (s i n 72^{o} + c o s 18^{o}) (c o s (90^{o} - 72^{o}) - c o s 18^{o}) = (s i n 72^{o} + c o s 18^{o}) (c o s 18^{o} - c o s 18^{o}) = (s i n 72^{o} + c o s 18^{o}) (0) (s i n 72^{o} + c o s 18^{o}) (0) = 0 = R H S$

(vi) $t a n 48^{\circ} t a n 23^{\circ} t a n 42^{\circ} t a n 67^{\circ} = 1 L H S = t a n 48^{o} t a n 23 v t a n 42^{o} t a n 67 v = c o t (90^{o} - 48^{o}) c o t (90^{o} - 23^{o}) t a n 42 v t a n 67^{o} = c o t 42^{o} c o t 67^{o} t a n 42^{o} t a n 67^{o} = \frac{1}{t a n 42^{o}} \times \frac{1}{t a n 67^{o}} \times t a n 42^{o} \times t a n 67^{o} = 1 = R H S$

Suggest Corrections

Similar questions

\sqrt{3}

(i) ${(\frac{s i n 49^{o}}{c o s 41^{o}})}^{2} + {(\frac{c o s 41^{o}}{s i n 49^{o}})}^{2}$

(ii) $c o s 48^{o} - s i n 42^{o}$

(iii) $\frac{c o t 40^{o}}{t a n 50^{o}} - \frac{1}{2} (\frac{c o s 35^{o}}{s i n 55^{o}})$

(iv) ${(\frac{s i n 27^{o}}{c o s 63^{o}})}^{2} + {(\frac{c o s 63^{o}}{s i n 27^{o}})}^{2}$

(v) $\frac{t a n 35^{o}}{t a n 55^{o}} + \frac{c o t 78^{o}}{t a n 12^{o}} - 1$

(vi) $\frac{s e c 70^{o}}{c o s e c 20^{o}} + \frac{s i n 59^{o}}{c o s 31^{o}}$

(vii) $c o s e c 31^{o} - s e c 59^{o}$

(viii) $(s i n 72^{o} + c o s 18^{o}) (s i n 72^{o} - c o s 18^{o})$

(ix) $s i n 35^{o} s i n 55^{o} - c o s 35^{o} c o s 55^{o}$

(x) tan $48^{o} t a n 23^{o} t a n 42^{o} t a n 67^{o}$

(xi) $s e c 50^{o} s i n 40^{o} + c o s 40^{o} c o s e c 50^{o}$

{(\frac{\sin 49 °}{\cos 41 °})}^{2} + {(\frac{\cos 41 °}{\sin 49 °})}^{2}

\frac{\cot 40 °}{\tan 50 °} - \frac{1}{2} (\frac{\cos 35 °}{\sin 55 °})

{(\frac{\sin 27 °}{\cos 63 °})}^{2} - {(\frac{\cos 63 °}{\sin 27 °})}^{2}

\frac{\tan 35 °}{\cot 55 °} + \frac{\cot 78 °}{\tan 12 °} - 1

\frac{\sec 70 °}{cosec 20 °} + \frac{\sin 59 °}{\cos 31 °}

Without using tables, show that (cos $35^{\circ}$ cos $55^{\circ}$ - sin $35^{\circ}$ sin $55^{\circ}$ ) = 0

Join BYJU'S Learning Program