(English Medium) ICSE Class 10 - CISCE Question Bank Solutions for Mathematics

Prove that  `sin^2 θ/ cos^2 θ + cos^2 θ/sin^2 θ = 1/(sin^2 θ. cos^2 θ) - 2`.

If x sin³θ + y cos³ θ = sin θ cos θ  and x sin θ = y cos θ , then show that x² + y² = 1.

Using ruler and compasses construct:
(i) a triangle ABC in which AB = 5.5 cm, BC = 3.4 cm and CA = 4.9 cm.
(ii) the locus of point equidistant from A and C.
(iii) a circle touching AB at A and passing through C.

Using only a ruler and compass construct ∠ABC = 120°, where AB = BC = 5 cm.
(i) Mark two points D and E which satisfy the condition that they are equidistant from both ABA and BC.
(ii) In the above figure, join AD, DC, AE and EC. Describe the figures:
(a) AECB, (b) ABD, (c) ABE.

Prove that `tan A/(1 + tan^2 A)^2 + cot A/(1 + cot^2 A)^2 = sin A.cos A`

Without using set squares or a protractor, construct:

1. Triangle ABC, in which AB = 5.5 cm, BC = 3.2 cm and CA = 4.8 cm.
2. Draw the locus of a point which moves so that it is always 2.5 cm from B.
3. Draw the locus of a point which moves so that it is equidistant from the sides BC and CA.
4. Mark the point of intersection of the loci with the letter P and measure PC.

Prove that `sin A/(sec A + tan A - 1) + cos A/(cosec A + cot A - 1) = 1`.

Prove that `tan^3 θ/( 1 + tan^2 θ) + cot^3 θ/(1 + cot^2 θ) = sec θ. cosec θ - 2 sin θ cos θ.`

Use ruler and compasses only for this question. Draw a circle of radius 4 cm and mark two chords, AB and AC, of the circle of length 6 cm and 5 cm respectively.

1. Construct the locus of points, inside the circle, that are equidistant from A and C. Prove your construction.
2. Construct the locus of points, inside the circle, that are equidistant from AB and AC.

Prove that `((1 + sin θ - cos θ)/( 1 + sin θ + cos θ))^2 = (1 - cos θ)/(1 + cos θ)`.

How will you find a point equidistant from three given points A, B, C which are not in the same straight line?

Without using set squares or protractor.
(i) Construct a ΔABC, given BC = 4 cm, angle B = 75° and CA = 6 cm.
(ii) Find the point P such that PB = PC and P is equidistant from the side BC and BA. Measure AP.

Prove that cos θ sin (90° - θ) + sin θ cos (90° - θ) = 1.

Prove that `cos θ/sin(90° - θ) + sin θ/cos (90° - θ) = 2`.

Prove that `(sin θ. cos (90° - θ) cos θ)/sin( 90° - θ) + (cos θ sin (90° - θ) sin θ)/(cos(90° - θ)) = 1`.

Proved that cosec²(90° - θ) - tan² θ = cos²(90° - θ)  +  cos² θ.

Without using the trigonometric table, prove that
tan 10° tan 15° tan 75° tan 80° = 1

Without using the trigonometric table, prove that
cos 1°cos 2°cos 3° ....cos 180° = 0.

Without using trigonometric table, prove that
`cos^2 26° + cos 64° sin 26° + (tan 36°)/(cot 54°) = 2`

Without using a trigonometric table, prove that
`(cos 70°)/(sin 20°) + (cos 59°)/(sin 31°) - 8sin^2 30° = 0`.