NCERT solutions for Mathematics Exemplar Class 10 chapter 8 - Introduction to Trignometry & its Equation [Latest edition]

If cos A = `4/5`, then the value of tan A is ______.

If sin A = `1/2`, then the value of cot A is ______.

The value of the expression [cosec(75° + θ) – sec(15° – θ) – tan(55° + θ) + cot(35° – θ)] is ______.

Given that sinθ = `a/b`, then cosθ is equal to ______.

If cos(α + β) = 0, then sin(α – β) can be reduced to ______.

The value of (tan1° tan2° tan3° ... tan89°) is ______.

If cos 9α = sinα and 9α < 90°, then the value of tan5α is ______.

If ∆ABC is right-angled at C, then the value of cos(A + B) is ______.

If sinA + sin²A = 1, then the value of the expression (cos²A + cos⁴A) is ______.

Given that sinα = `1/2` and cosβ = `1/2`, then the value of (α + β) is ______.

The value of the expression `[(sin^2 22^circ + sin^2 68^circ)/(cos^2 22^circ + cos^2 68^circ) + sin^2  63^circ + cos 63^circ sin 27^circ]` is ______.

If 4 tanθ = 3, then `((4 sintheta - costheta)/(4sintheta + costheta))` is equal to ______.

If sinθ – cosθ = 0, then the value of (sin⁴θ + cos⁴θ) is ______.

sin(45° + θ) – cos(45° – θ) is equal to ______.

A pole 6 m high casts a shadow `2sqrt(3)` m long on the ground, then the Sun’s elevation is ______.

`tan 47^circ/cot 43^circ` = 1

The value of the expression (cos² 23° – sin² 67°) is positive.

The value of the expression (sin 80° – cos 80°) is negative.

`sqrt((1 - cos^2theta) sec^2 theta) = tan theta` 

If cosA + cos²A = 1, then sin²A + sin⁴A= 1.

(tan θ + 2)(2 tan θ + 1) = 5 tan θ + sec²θ.

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

If a man standing on a platform 3 metres above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is equal to the angle of depression of its reflection.

The value of 2sinθ can be `a + 1/a`, where a is a positive number, and a ≠ 1.

Write True' or False' and justify your answer the following : 

\[ \cos \theta = \frac{a^2 + b^2}{2ab}\]where a and b are two distinct numbers such that ab > 0.

The angle of elevation of the top of a tower is 30°. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.

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

Prove the following:

`sintheta/(1 + cos theta) + (1 + cos theta)/sintheta` = 2cosecθ

Prove the following:

`tanA/(1 + sec A) - tanA/(1 - sec A)` = 2cosec A

Prove the following:

If tan A = `3/4`, then sinA cosA = `12/25`

Prove the following:

(sin α + cos α)(tan α + cot α) = sec α + cosec α

Prove the following:

`(sqrt(3) + 1) (3 - cot 30^circ)` = tan³ 60° – 2 sin 60°

Prove the following

`1 + (cot^2 alpha)/(1 + cosec alpha)` = cosec α

Prove the following:

tanθ + tan (90° – θ) = sec θ sec(90° – θ)

Prove the following:

Find the angle of elevation of the sun when the shadow of a pole h metres high is `sqrt(3)` h metres long.

Prove the following:

If `sqrt(3) tan theta = 1`, then find the value of `sin^2 theta - cos^2 theta`.

Prove the following:

A ladder 15 metres long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, find the height of the wall.

Prove the following:

Simplify (1 + tan²θ) (1 – sinθ)(1 + sinθ)

Prove the following:

If 2sin²θ – cos²θ = 2, then find the value of θ.

Prove the following:

Show that `(cos^2 (45^circ + theta) + cos^2 (45^circ - theta))/(tan(60^circ + theta) tan(30^circ - theta))` = 1

An observer 1.5 metres tall is 20.5 metres away from a tower 22 metres high. Determine the angle of elevation of the top of the tower from the eye of the observer.

Show that tan⁴θ + tan²θ = sec⁴θ – sec²θ.

If cosecθ + cotθ = p, then prove that cosθ = `(P^2 - 1)/(p^2 + 1)`

Prove that `sqrt(sec^2 theta + cosec^2 theta) = tan theta + cot theta`

The angle of elevation of the top of a tower from certain point is 30°. If the observer moves 20 metres towards the tower, the angle of elevation of the top increases by 15°. Find the height of the tower.

If 1 + sin²θ = 3sinθ cosθ , then prove that tanθ = 1 or `1/2`.

Given that sinθ + 2cosθ = 1, then prove that 2sinθ – cosθ = 2.

The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. Prove that the height of the tower is `sqrt(st)`

The shadow of a tower standing on a level plane is found to be 50 m longer when Sun’s elevation is 30° than when it is 60°. Find the height of the tower.

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. Prove that the height of the tower is `((h tan alpha)/(tan beta - tan alpha))`

If tan θ + sec θ = l, then prove that secθ = `(l^2 + 1)/(2l)`

If sinθ + cosθ = p and sec θ + cosec θ = q, then prove that q(p² – 1) = 2p.

If a sinθ + b cosθ = c, then prove that a cosθ – b sinθ = `sqrt(a^2 + b^2 - c^2)`

Prove that `(1 + sec theta - tan theta)/(1 + sec theta + tan theta) = (1 - sin theta)/cos theta`

The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60° and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. Find the distance between the two towers and also the height of the other tower.

From the top of a tower h m high, the angles of depression of two objects, which are in line with the foot of the tower are α and β (β > α). Find the distance between the two objects.

A ladder rests against a vertical wall at an inclination α to the horizontal. Its foot is pulled away from the wall through a distance p so that its upper end slides a distance q down the wall and then the ladder makes an angle β to the horizontal. Show that `p/q = (cos beta - cos alpha)/(sin alpha - sin beta)`

The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 45°. Find the height of the tower.

A window of a house is h metre above the ground . From the window , the angles of elevation and depression of the top and bottom of another house situated on the opposite side of the lane are found to be α and β respectively. Prove that the height of the house is h(1+ tan α tan β) metres.

The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window. At certain instant the angles of elevation of a balloon from these windows are observed to be 60° and 30°, respectively. Find the height of the balloon above the ground.