Exercise 8.3
निम्नलिखित के मान ज्ञात कीजिए ।
Question 31
$\frac{\sin 50^{\circ}}{\cos 40^{\circ}}+\frac{\operatorname{cosec}
40^{\circ}}{\sec 50^{\circ}}-4 \cos 50^{\circ} \cdot \operatorname{cosec}
40^{\circ}$
Sol :
$\frac{\cos \left(90^{\circ}-50^{\circ}\right)}{\cos 40^{\circ}}+\frac{\sec \left(90^{\circ}-40^{\circ}\right)}{\sec 50^{\circ}}-4 \sin \left(90^{\circ}-50^{\circ}\right) \cdot \operatorname{Cosec} 40^{\circ}$
$\frac{\cos 40^{\circ}}{\cos 40^{\circ}}+\frac{\sec 50^{\circ}}{\sec 50^{\circ}}-4 \sin 40^{\circ} \cdot \operatorname{Cosec} 40^{\circ}$
=1+1-4×1
=2-4=2
Question 32
$\frac{\cos ^{2} 20^{\circ}+\cos ^{2} 70^{\circ}}{\sin ^{2}
59^{\circ}+\sin ^{2} 31^{\circ}}+\sin 35^{\circ} \cdot \sec 55^{\circ}$
Sol :
$\frac{\sin ^{2}\left(90^{\circ}-20^{\circ}\right)+\cos ^{2} 70^{\circ}}{\cos ^{2}\left(90^{\circ}-59^{\circ}\right)+\sin ^{2} 31^{\circ}}+\cos \left(90^{\circ}-35^{\circ}\right) \cdot \operatorname{Sec} 55^{\circ}$
$\frac{\sin ^{2} 70^{\circ}+\cos ^{2} 70^{\circ}}{\cos ^{2} 31^{\circ}+\sin ^{2} 31^{\circ}}+\cos 55^{\circ} \cdot \operatorname{Sec} 55^{\circ}$
$\frac{1}{1}+1=2$
Question 33
$\frac{\tan 50^{\circ}+\sec 50^{\circ}}{\cot
40^{\circ}+\operatorname{cosec} 40^{\circ}}+\cos 40^{\circ} \cdot
\operatorname{cosec} 50^{\circ}$
Sol :
$\frac{\cot \left(90^{\circ}-5^{\circ}\right)+\operatorname{cose}\left(90^{\circ}-50^{\circ}\right)}{\cot 40^{\circ}+\operatorname{cosec} 40^{\circ}}+\sin \left(90^{\circ}-40^{\circ}\right) \cdot \operatorname{cosec} 50^{\circ}$
$\frac{\cot 40^{\circ}+\operatorname{cosec} 40^{\circ}}{\cot 40^{\circ}+\operatorname{cosec} 40^{\circ}}+\sin 50^{\circ} \cdot \operatorname{cosec} 50^{\circ}$
1+1=2
Question 34
$\operatorname{cosec}\left(65^{\circ}+\theta\right)-\sec
\left(25^{\circ}-\theta\right)-\tan \left(55^{\circ}-\theta\right)+\cot
\left(35^{\circ}+\theta\right)$
Sol :
$\sec \left(90^{\circ}-\left(65^{\circ}+\theta\right)\right)-\sec \left(25^{\circ}-\theta\right)-\tan \left(55^{\circ}-\theta\right)+\cot \left(90^{\circ}-\left(35^{\circ}+\theta\right)\right)$
$\sec \left(90^{\circ}-65^{\circ}-\theta\right)-\sec \left(25^{\circ}-\theta\right)-\tan \left(55^{\circ}-\theta\right)+\cot \left(90^{\circ}-35^{\circ}-\theta\right)$
$\sec \left(25^{\circ}-\theta\right)-\sec \left(25^{\circ}-\theta\right)-\tan \left(55^{\circ}-\theta\right)+\cot \left(55^{\circ}-\theta\right)$
=0
Question 35
$\frac{\cos 35^{\circ}}{\sin 55^{\circ}}+\frac{\sin 11^{\circ}}{\cos
79^{\circ}}-\cos 28^{\circ} \cdot \operatorname{cosec} 62^{\circ}$
Sol :
$\frac{\sin \left(90^{\circ}-3^{\circ}\right)}{\sin 55^{\circ}}+\frac{\cos \left(90^{\circ}-11^{\circ}\right)}{\cos 79^{\circ}}-\sin \left(90^{\circ}-28^{\circ}\right) . \operatorname{Cosec} 62^{\circ}$
$\frac{\sin 55^{\circ}}{\sin 55^{\circ}}+\frac{\cos 79^{\circ}}{\cos 79^{\circ}}-\sin \left(90^{\circ}-28^{\circ}\right) \cdot \operatorname{Cosec} 62^{\circ}$
1+1-1=2-1=1
Question 36
$\frac{\cos ^{2} 20^{\circ}+\cos ^{2} 70^{\circ}}{\sin ^{2}
59^{\circ}+\sin ^{2} 31^{\circ}}$
Sol :
$\frac{\sin ^{2}\left(90^{\circ}-20^{\circ}\right)+\cos ^{2} 70^{\circ}}{\cos ^{2}\left(90^{\circ}-59^{\circ}\right)+\sin ^{2} 31^{\circ}}$
$\frac{\sin ^{2} 70^{\circ}+\cos ^{2} 70^{\circ}}{\cos ^{2} 31^{\circ}+\sin ^{2} 31^{\circ}}$
$\frac{1}{1}=1$
Question 37
$\operatorname{cosec}\left(65^{\circ}+, \theta\right)-\sec
\left(25^{\circ}-\theta\right)$
Sol :
$\sec \left(90^{\circ}-\left(65^{\circ}+\theta\right)\right)-\sec \left(25^{\circ}-\theta\right)$
$\sec \left(90^{\circ}-65^{\circ}-\theta\right)-\sec \left(25^{\circ}-\theta\right)$
$\sec \left(25^{\circ}-\theta\right)-\sec \left(25^{\circ}-\theta\right)$
=0
Question 38
$\cos \left(60^{\circ}+\theta\right)-\sin \left(30^{\circ}-\theta\right)$
Sol :
$\sin \left(90^{\circ}-\left(60^{\circ}+\theta\right)\right)-\sin \left(30^{\circ}-\theta\right)$
$\sin \left(90^{\circ}-60^{\circ}-\theta\right)-\sin \left(30^{\circ}-\theta\right)$
$\sin \left(30^{\circ}-\theta\right)-\sin \left(30^{\circ}-\theta\right)=0$
Question 39
$\sec 70^{\circ} \cdot \sin 20^{\circ}-\cos 20^{\circ} \cdot
\operatorname{cosec} 70^{\circ}$
Sol :
$\operatorname{cosec}\left(90^{\circ}-70^{\circ}\right) \cdot \sin 20^{\circ} \cdot-\sin \left(90^{\circ}-20^{\circ}\right) \cdot \operatorname{cosec} 70^{\circ}$ $\operatorname{cosec} 20^{\circ} . \sin 20^{\circ} .-\sin 70^{\circ} . \operatorname{cosec} 70^{\circ}$
1-1=0
Question 40
$\left(\sin 72^{\circ}+\cos 18^{\circ}\right)\left(\sin 72^{\circ}-\cos
18^{\circ}\right)$
Sol :
$=\sin ^{2} 72^{\circ}-\cos ^{2} 18^{\circ}$
$=\cos ^{2}\left(90^{\circ}-72^{\circ}\right)-\cos ^{2} 18^{\circ}$
$=\cos ^{2} 18^{\circ}-\cos ^{2} 18^{\circ}$
=0
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