NCERT Exemplar Solutions
Class 10 - Mathematics

CHAPTER 8: Introduction to Trignometry and Its Applications

\(\dfrac{\tan 47^\circ}{\cot 43^\circ}=1\). Write ‘True’ or ‘False’ and justify.

The value of the expression \(\cos^2 23^\circ-\sin^2 67^\circ\) is positive.

The value of the expression \(\sin 80^\circ-\cos 80^\circ\) is negative.

\(\sqrt{1-\cos^2\theta}\;\sec^2\theta=\tan\theta\).

If \(\cos A+\cos^2A=1\), then \(\sin^2A+\sin^4A=1\).

\((\tan\theta+2)(2\tan\theta+1)=5\tan\theta+\sec^2\theta\).

If the length of the shadow of a tower is increasing, then the angle of elevation of the sun is also increasing.

A man 3 m above the surface of a lake observes a cloud and its reflection in the lake. The angle of elevation of the cloud equals the angle of depression of its reflection.

The value of \(2\sin\theta\) can be \(a+\dfrac{1}{a}\), where \(a\gt0\) and \(a\ne1\).

\(\cos\theta=\dfrac{a^2+b^2}{2ab}\), where \(a,b\) are distinct and \(ab\gt0\).

The angle of elevation of the top of a tower is \(30^\circ\). If the height of the tower is doubled (observer fixed), then the angle of elevation also doubles.

If both the height of a tower and the distance of the point of observation from its foot increase by 10%, then the angle of elevation of its top remains unchanged.

Prove that \(\dfrac{\sin\theta}{1+\cos\theta}+\dfrac{1+\cos\theta}{\sin\theta}=2\csc\theta\).

Prove that \(\dfrac{\tan A}{1+\sec A}-\dfrac{\tan A}{1-\sec A}=2\csc A\).

If \(\tan A=\dfrac{3}{4}\), prove that \(\sin A\cos A=\dfrac{12}{25}\).

Prove that \((\sin\alpha+\cos\alpha)(\tan\alpha+\cot\alpha)=\sec\alpha+\csc\alpha\).

Prove that \((\sqrt{3}+1)(3-\cot 30^\circ)=\tan^3 60^\circ-2\sin 60^\circ\).

Prove that \(1+\dfrac{\cot^2\alpha}{1+\csc\alpha}=\cosec\alpha\).

Prove that \(\tan\theta+\tan(90^\circ-\theta)=\sec\theta\,\sec(90^\circ-\theta)\).

The shadow of a pole of height \(h\) metres is \(\sqrt{3}\,h\) metres long. Find the angle of elevation of the Sun.

If \(\sqrt{3}\,\tan\theta=1\), find \(\sin^2\theta-\cos^2\theta\).

A \(15\) m long ladder just reaches the top of a vertical wall making an angle of \(60^\circ\) with the wall. Find the height of the wall.

Simplify \((1+\tan^2\theta)(1-\sin\theta)(1+\sin\theta)\).

If \(2\sin^2\theta-\cos^2\theta=2\), find \(\theta\).

Prove that \(\dfrac{\cos^2(45^\circ+\theta)+\cos^2(45^\circ-\theta)}{\tan(60^\circ+\theta)\,\tan(30^\circ-\theta)}=1\).

An observer \(1.5\) m tall is \(20.5\) m from a tower \(22\) m high. Find the angle of elevation of the top of the tower from the observer's eye.

Prove that \(\tan^4\theta+\tan^2\theta=\sec^4\theta-\sec^2\theta\).

If \(\csc\theta + \cot\theta = p\), prove that \(\cos\theta = \dfrac{p^2-1}{p^2+1}\).

Prove that \(\sqrt{\sec^2\theta+\csc^2\theta}=\tan\theta+\cot\theta\) for acute \(\theta\).

The angle of elevation of the top of a tower from a point is \(30^\circ\). If the observer moves \(20\,\text{m}\) towards the tower, the angle becomes \(45^\circ\). Find the height.

If \(1+\sin^2\theta=3\sin\theta\cos\theta\), prove that \(\tan\theta=1\) or \(\dfrac12\).

Given \(\sin\theta+2\cos\theta=1\), show that \(|2\sin\theta-\cos\theta|=2\).

The angles of elevation of the top of a tower from two points \(s\) and \(t\) metres from its foot are complementary. Prove that the height is \(\sqrt{st}\).

The shadow of a tower is \(50\,\text{m}\) longer when the Sun’s elevation is \(30^\circ\) than when it is \(60^\circ\). Find the height.

A tower of unknown height is surmounted by a vertical flag staff of height \(h\). At a point, the angles of elevation of the bottom and the top of the flag staff are \(\alpha\) and \(\beta\). Prove that the height of the tower is \(\dfrac{h\tan\alpha}{\tan\beta-\tan\alpha}\).

If \(\tan\theta+\sec\theta=\ell\), prove that \(\sec\theta=\dfrac{\ell^2+1}{2\ell}\).

If \(\sin\theta+\cos\theta=p\) and \(\sec\theta+\csc\theta=q\), prove that \(q(p^2-1)=2p\).

If \(a\sin\theta+b\cos\theta=c\), prove that \(a\cos\theta-b\sin\theta=\pm\sqrt{a^2+b^2-c^2}\).

Prove that \(\dfrac{1+\sec\theta-\tan\theta}{1+\sec\theta+\tan\theta}=\dfrac{1-\sin\theta}{\cos\theta}\).

Two towers stand on the same plane. From the foot of the second, the angle of elevation of the top of the first (height \(30\,\text{m}\)) is \(60^\circ\). From the foot of the first, the angle of elevation of the top of the second is \(30^\circ\). Find the distance between the towers and the height of the second.

From the top of a tower of height \(h\), the angles of depression of two objects in line with the foot are \(\alpha\) and \(\beta\) (with \(\beta>\alpha\)). Find the distance between the objects.

A ladder rests against a wall at angle \(\alpha\). Its foot is pulled away by \(p\) metres so that its top slides down \(q\) metres and now makes angle \(\beta\) with the ground. Prove that

\[\dfrac{p}{q}=\dfrac{\cos\beta-\cos\alpha}{\sin\alpha-\sin\beta}.\]

From a point on the ground the elevation of a tower is \(60^\circ\). From another point \(10\,\text{m}\) vertically above the first, the elevation is \(45^\circ\). Find the height of the tower.

From a window at height \(h\) the angles of elevation and depression of the top and the bottom of another house are \(\alpha\) and \(\beta\), respectively. Prove that the height of the other house is \(h\big(1+\tan\alpha\,\cot\beta\big)\).

The lower window of a house is at \(2\,\text{m}\) above the ground and the upper window is \(4\,\text{m}\) vertically above it. The angles of elevation of a balloon from these windows are \(60^\circ\) and \(30^\circ\), respectively. Find the height of the balloon above the ground.