Selina solutions concise maths class 9 icse Solutions for Chapter 25 chapter 25 complementary angles

Evaluate:

\begin{aligned} &\text { (i) }\\ &\frac{\cos 22^{\circ}}{\sin 68^{\circ}} \end{aligned}

\text { (ii) } \frac{\tan 47^{\circ}}{\cot 43^{\circ}}

\text { (iii) } \frac{\sec 75^{\circ}}{\operatorname{cosec} 15^{\circ}}

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

\begin{aligned} &(\mathbf{v})\\ &\sin ^{2} 40^{\circ}-\cos ^{2} 50^{\circ} \end{aligned}

\begin{aligned} &\text { (vi) }\\ &\sec ^{2} 18^{\circ}-\operatorname{cosec}^{2} 72^{\circ} \end{aligned}

\text { (vii) } \sin 15^{\circ} \cos 15^{\circ}-\cos 75^{\circ} \sin 75^{\circ}

\text { (viii) } \quad \sin 42^{\circ} \sin 48^{\circ}-\cos 42^{\circ} \cos 48^{\circ}

Evaluate:

(i) \sin \left(90^{\circ}-\mathrm{A}\right) \sin \mathrm{A}-\cos \left(90^{\circ}-\mathrm{A}\right) \cos \mathrm{A}

(ii) \sin ^{2} 35^{\circ}-\cos ^{2} 55^{\circ}

(iii) \frac{\cot 54^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\cot 70^{\circ}}-2

(iv) \frac{2 \tan 53^{\circ}}{\cot 37^{\circ}}-\frac{\cot 80^{\circ}}{\tan 10^{\circ}}

(v) \cos ^{2} 25^{\circ}-\sin ^{2} 65^{\circ}-\tan ^{2} 45^{\circ}

(vi) \left(\frac{\sin 77^{\circ}}{\cos 13^{\circ}}\right)^{2}+\left(\frac{\cos 77^{\circ}}{\sin 13^{\circ}}\right)^{2}-2 \cos ^{2} 45^{\circ}

Show that

(i) \tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}=1

(ii) \sin 42^{\circ} \sec 48^{\circ}+\cos 42^{\circ} \operatorname{cosec} 48^{\circ}=2

Express each of the following in terms of angles between 0^{0} \text { and } 45^{0}:

(i) \sin 59^{0}+\tan 63^{\circ}

(ii) \operatorname{cosec} 68^{0}+\cot 72^{\circ}

(iii) \cos 74^{0}+\sec 67^{0}

For triangle ABC, show that:

(i) \sin \left(\frac{A+B}{2}\right)=\cos \frac{C}{2}

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

Evaluate:

\begin{aligned} &\text { (i) }\\ &3 \frac{\sin 72^{\circ}}{\cos 18^{\circ}}-\frac{\sec 32^{\circ}}{\operatorname{cosec} 58^{\circ}} \end{aligned}

\text { (ii) } \quad 3 \cos 80^{\circ} \operatorname{cosec} 10^{\circ}+2 \sin 59^{\circ} \sec 31^{0}

\begin{aligned} &\text { (iii) }\\ &\frac{\sin 80^{\circ}}{\cos 10^{\circ}}+\sin 59^{\circ} \sec 31^{0} \end{aligned}

\text { (iv) } \tan \left(55^{\circ}-\mathrm{A}\right)-\cot \left(35^{0}+\mathrm{A}\right)

\text { (v) } \operatorname{cosec}\left(65^{0}+A\right)-\sec \left(25^{0}-A\right)

\text { (vi) } 2 \frac{\tan 57^{\circ}}{\cot 33^{\circ}}-\frac{\cot 70^{\circ}}{\tan 20^{\circ}}-\sqrt{2} \cos 45^{\circ}

\begin{aligned} &\text { (vii) }\\ &\frac{\cot ^{2} 41^{0}}{\tan ^{2} 49^{\circ}}-2 \frac{\sin ^{2} 75^{\circ}}{\cos ^{2} 15^{\circ}} \end{aligned}

\text { (viii) } \frac{\cos 70^{\circ}}{\sin 20^{\circ}}+\frac{\cos 59^{\circ}}{\sin 31^{\circ}}-8 \sin ^{2} 30^{\circ}

\text { (ix) } 14 \sin 30^{\circ}+6 \cos 60^{\circ}-5 \tan 45^{\circ}

A triangle ABC is right angled at B.

Find the value of

\frac{\sec A \cdot \sin C-\tan A \cdot \tan C}{\sin B}

In each case, given below, find the value of angle A, where 0^{\circ} \leq \mathbf{A} \leq 90^{\circ}.

\sin \left(90^{\circ}-3 \mathrm{A}\right) \cdot \operatorname{cosec} 42^{\circ}=1

(i) \sin \left(90^{\circ}-3 \mathrm{A}\right) \cdot \operatorname{cosec} 42^{\circ}=1

(ii) \cos \left(90^{\circ}-3 \mathrm{A}\right) \cdot \sec 77^{\circ}=1