R s aggarwal solutions mathematics class 10 Solutions for Chapter 9 trigonometric ratios of complementary angles

Without using trigonometric tables, evaluate:

\text { (i) } \frac{\sin 16^{\circ}}{\cos 74^{\circ}}

\text { (ii) } \frac{\sec 11^{0}}{\operatorname{cosec} 79^{\circ}}

\text { (iii) } \frac{\tan 27^{\circ}}{\cot 63^{\circ}}

\text { (iv) } \frac{\cos 35^{\circ}}{\sin 55^{\circ}}

\text { (v) } \frac{\operatorname{cosec} 42^{\circ}}{\sec 48^{\circ}}

\text { (vi) } \frac{\cot 38^{\circ}}{\tan 52^{\circ}}

Without using trigonometric tables, prove that:

(i) cos 81° − sin 9° = 0

(ii) tan 71° − cot 19° = 0

(iii) cosec 80° − sec 10° = 0

(iv) cosec2 72° − tan2 18° = 1

(v) cos2 75° + cos2 15° = 1

(vi) tan2 66° − cot2 24° = 0

(vii) sin2 48° + sin2 42° = 1

(viii) cos2 57° − sin2 33° = 0

(ix) (sin 65° + cos 25°)(sin 65° − cos 25°) = 0

Prove that:

\text { (i) } \frac{\sin 70^{\circ}}{\cos 20^{\circ}}+\frac{\operatorname{cosec} 20^{\circ}}{\sec 70^{\circ}}-2 \cos 70^{\circ} \operatorname{cosec} 20^{\circ}=0

\text { (ii) } \frac{\cos 80^{\circ}}{\sin 10^{\circ}}+\cos 59^{\circ} \operatorname{cosec} 31^{\circ}=2

\text { (iii) } \frac{2 \sin 68^{\circ}}{\cos 22^{\circ}}-\frac{2 \cot 15^{\circ}}{5 \tan 75^{\circ}}-\frac{3 \tan 45^{\circ} \tan 20^{\circ} \tan 40^{\circ} \tan 50^{\circ} \tan 70^{\circ}}{5}=1

\text { (iv) } \frac{\sin 18^{\circ}}{\cos 72^{\circ}}+\sqrt{3}\left(\tan 10^{\circ} \tan 30^{\circ} \tan 40^{\circ} \tan 50^{\circ} \tan 80^{\circ}\right)=2

(\mathrm{v}) \frac{7 \cos 55^{\circ}}{3 \sin 35^{\circ}}-\frac{4\left(\cos 70^{\circ} \operatorname{cosec} 20^{\circ}\right)}{3\left(\tan 5^{\circ} \tan 25^{\circ} \tan 45^{\circ} \tan 65^{\circ} \tan 85^{\circ}\right)}=1

Prove that:

(i) \sin \theta \cos \left(90^{\circ}-\theta\right)+\sin \left(90^{\circ}-\theta\right) \cos \theta=1

(ii) \frac{\sin \theta}{\cos (90-\theta)}+\frac{\cos \theta}{\sin (90-\theta)}=2

(iii)

\frac{\sin \theta \cos \left(90^{\circ}-\theta\right) \cos \theta}{\sin \left(90^{\circ}-\theta\right)}+\frac{\cos \theta \sin \left(90^{\circ}-\theta\right) \sin \theta}{\cos \left(90^{\circ}-\theta\right)}=1

\text { (iv) } \frac{\cos \left(90^{\circ}-\theta\right) \sec \left(90^{\circ}-\theta\right) \tan \theta}{\operatorname{cosec}\left(90^{\circ}-\theta\right) \sin \left(90^{\circ}-\theta\right) \cot \left(90^{\circ}-\theta\right)}+\frac{\tan \left(90^{\circ}-\theta\right)}{\cot \theta}=2

\text { (v) } \frac{\cos \left(90^{\circ}-\theta\right)}{1+\sin \left(90^{\circ}-\theta\right)}+\frac{1+\sin \left(90^{\circ}-\theta\right)}{\cos \left(90^{\circ}-\theta\right)}=2 \operatorname{cosec} \theta

\text { (vi) } \frac{\sec \left(90^{\circ}-\theta\right) \operatorname{cosec} \theta-\tan \left(90^{\circ}-\theta\right) \cot \theta+\cos ^{2} 25^{\circ}+\cos ^{2} 65^{\circ}}{3 \tan 27^{\circ} \tan 63^{\circ}}=\frac{2}{3}

\begin{aligned} &\text { (vii) }\\ &\cot \theta \tan \left(90^{\circ}-\theta\right)-\sec \left(90^{\circ}-\theta\right) \operatorname{cosec} \theta+\sqrt{3} \tan 12^{\circ} \tan 60^{\circ} \tan 78^{\circ}=2 \end{aligned}

Prove that:

(i) tan 5° tan25° tan30° tan65° tan85° = 1/√3

(ii) cot 12° cot38° cot52° cot60° cot78° = 1/√3

(iii) cos 15° cos35° cosec55° cos60° cosec75° = 1/2

(iv) cos 1° cos2° cos3° … cos180° = 0

(v)\left(\frac{\sin 49^{0}}{\cos 41^{0}}\right)^{2}+\left(\frac{\cos 41^{0}}{\sin 49^{0}}\right)^{2}=2

Prove that:

(i) sin (70° + θ) − cos (20° − θ) = 0

(ii) tan (55° − θ) − cot (35° + θ) = 0

(iii) cosec (67° + θ) − sec (23° − θ) = 0

(iv) cosec (65 °+ θ) sec (25° − θ) − tan (55° − θ) + cot (35° + θ) = 0

(v) sin (50° + θ ) − cos (40° − θ) + tan 1° tan 10° tan 80° tan 89° = 1.

Express each of the following in terms of trigonometric ratios of angles lying between 0° and 45°.

(i) sin 67° + cos 75°

(ii) cot 65° + tan49°

(iii) sec78° + cosec56°

(iv) cosec54° + sin72°

If A, B and C are the angles of an ∆ABC, prove that:

\tan \left(\frac{C+A}{2}\right)=\cot \frac{B}{2}

If cos2θ = sin4θ, where 2θ and 4θ are acute angles, then find the value of θ.

If sec2A = cosec(A − 42°), where 2A is an acute angle, then find the value of A.

If sin 3 A = cos (A − 26°), where 3 A is an acute angle, find the value of A.

If tan 2 A = cot (A − 12°), where 2 A is an acute angle, find the value of A.

If sec 4 A = cosec (A − 15°), where 4 A is an acute angle, find the value of A.

Prove that:

\frac{2}{3} \operatorname{cosec}^{2} 58^{\circ}-\frac{2}{3} \cot 58^{\circ} \tan 32^{\circ}-\frac{5}{3} \tan 13^{\circ} \tan 37^{\circ} \tan 45^{\circ} \tan 53^{\circ} \tan 77^{\circ}=-1