KC Sinha Mathematics Solution Class 10 Chapter 8 त्रिकोणमितीय अनुपात एवम सर्वसमिकाए ( Trigonometry Ratios and Identities ) Exercise 8.3 (Q31-Q40)

 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