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In any ΔABC, prove that

a(b cos C – c cos B) = (b^{2} – c^{2})

ac cos B – bc cos A = (a^{2} – b^{2})

2(bc cos A + ca cos B + ab cos C) = (a^{2} + b^{2} + c^{2})

a sin A – b sin B = c sin (A – B)

a^{2} sin ( B – C ) = (b^{2} – c^{2}) sin A

a^{2}(cos^{2}B – cos^{2}C) + b^{2}(cos^{2}C – cos^{2}A) + c^{2}(cos^{2}A – cos^{2}B) = 0

(c^{2} – a^{2} + b^{2}) tan A = (a^{2} – b^{2} + c^{2}) tan B = (b^{2} – c^{2} + a^{2}) tan C

If in a ∆ABC, ∠C = 90^{0}, then prove that .

In a ∆ABC, if , show that the triangle is isosceles.

In a ∆ABC, if , show that the triangle is right-angled.

Solve the triangle in which a = 2 cm, b = 1 cm and c = cm.

In a ΔABC, if a = 3 cm, b = 5 cm and c = 7 cm, find cos A, cos B, cos C.

If the angles of a triangle are in the ratio 1 : 2 : 3, prove that its corresponding sides are in the ratio .

Two boats leave a port at the same time. One travels 60 km in the direction N 50^{0} E while the other travels 50 km in the direction S 70^{0} E. What is the distance between the boats?

A town B is 12 km south and 18 km west of a town A. Show that the bearing of B from A is S 56^{0} 20’ W. Also, find the distance of B from A.

At the foot of a mountain, the angle of elevation of its summit is 45^{0}. After ascending 1 km towards the mountain up an incline of 30^{0}, the elevation changes to 60^{0} (as shown in the given figure). Find the height of the mountain. [Given : ]