Educational

Prove the Identity : \( \cosec x(\sec x-1)-\cot x(1-\cos x)=\tan x-\sin x \) Solution: \[ \cosec x(\sec x-1)-\cot x(1-\cos x) \] \[ =\frac{1}{\sin x}\left(\frac{1}{\cos x}-1\right) -\frac{\cos x}{\sin x}(1-\cos x) \] \[ =\frac{1-\cos x}{\sin x\cos x} -\frac{\cos x(1-\cos x)}{\sin x} \] \[ =\frac{1-\cos x-\cos^2 x(1-\cos x)}{\sin x\cos x} \] \[ =\frac{(1-\cos x)(1-\cos^2 x)}{\sin x\cos x} \] […]

Prove the Identity : \( (\cosec x-\sin x)(\sec x-\cos x)(\tan x+\cot x)=1 \) Solution: \[ (\cosec x-\sin x)(\sec x-\cos x)(\tan x+\cot x) \] \[ =\left(\frac{1}{\sin x}-\sin x\right) \left(\frac{1}{\cos x}-\cos x\right) \left(\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}\right) \] \[ =\frac{1-\sin^2 x}{\sin x}\cdot \frac{1-\cos^2 x}{\cos x}\cdot \frac{\sin^2 x+\cos^2 x}{\sin x\cos x} \] \[ =\frac{\cos^2 x}{\sin x}\cdot \frac{\sin^2 x}{\cos

Prove the Identity : \( \sin^6 x + \cos^6 x = 1 – 3\sin^2 x \cos^2 x \) Solution: \[ \sin^6 x + \cos^6 x \] \[ = (\sin^2 x)^3 + (\cos^2 x)^3 \] \[ = (\sin^2 x+\cos^2 x)(\sin^4 x-\sin^2 x\cos^2 x+\cos^4 x) \] \[ = \sin^4 x-\sin^2 x\cos^2 x+\cos^4 x \] \[ = (\sin^2

Prove the Identity : \( \sec^4 x – \sec^2 x = \tan^4 x + \tan^2 x \) Solution: \[ \sec^4 x – \sec^2 x \] \[ = \sec^2 x(\sec^2 x-1) \] \[ = \sec^2 x \tan^2 x \] \[ = (1+\tan^2 x)\tan^2 x \] \[ = \tan^2 x+\tan^4 x \] \[ = \tan^4 x+\tan^2 x

The Radius of the Circle Whose Arc of Length \(15\pi\) cm Makes an Angle of \(\frac{3\pi}{4}\) Radian at the Centre Question: The radius of the circle whose arc of length \(15\pi\) cm makes an angle of \(\frac{3\pi}{4}\) radian at the centre is (a) \(10\) cm (b) \(20\) cm (c) \(11 \frac{1}{4}\) cm (d) \(22 \frac{1}{2}\)

A Circular Wire of Radius 7 cm is Cut and Bent Again into an Arc of a Circle of Radius 12 cm Question: A circular wire of radius \(7\) cm is cut and bent again into an arc of a circle of radius \(12\) cm. The angle subtended by the arc at the center is

If OP Makes 4 Revolutions in One Second, the Angular Velocity in Radians per Second is Question: If OP makes \(4\) revolutions in one second, the angular velocity in radians per second is (a) \(\pi\) (b) \(2\pi\) (c) \(4\pi\) (d) \(8\pi\) Solution We know that: \[ 1 \text{ revolution} = 2\pi \text{ radians} \] Given

If the Arcs of the Same Length in Two Circles Subtend Angles 65° and 110° at the Centre, Then the Ratio of the Radii of the Circles is Question: If the arcs of the same length in two circles subtend angles \(65^\circ\) and \(110^\circ\) at the centre, then the ratio of the radii of the

At 3:40, the Hour and Minute Hands of a Clock are Inclined at Question: At \(3:40\), the hour and minute hands of a clock are inclined at (a) \(\frac{2\pi}{3}^c\) (b) \(\frac{7\pi}{12}^c\) (c) \(\frac{13\pi}{18}^c\) (d) \(\frac{3\pi}{4}^c\) Solution At \(3:40\), Minute hand position: \[ 40 \times 6^\circ = 240^\circ \] Hour hand position: \[ 3 \times 30^\circ

The Angle Between the Minute and Hour Hands of a Clock at 8:30 is Question: The angle between the minute and hour hands of a clock at 8:30 is (a) \(80^\circ\) (b) \(75^\circ\) (c) \(60^\circ\) (d) \(105^\circ\) Solution At \(8:30\), Minute hand position: \[ 30 \times 6^\circ = 180^\circ \] Hour hand position: \[ 8