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

If tan α = n tan β, sin α = m sin β, prove that cos² α  = `(m^2 - 1)/(n^2 - 1)`.

If cosθ + sinθ = `sqrt2` cosθ, show that cosθ - sinθ = `sqrt2` sinθ.

Without using set squares or protractor construct a triangle ABC in which AB = 4 cm, BC = 5 cm and ∠ABC = 120°.
(i) Locate the point P such that ∠BAp = 90° and BP = CP.
(ii) Measure the length of BP.

Prove that (cosec A - sin A)( sec A - cos A) sec² A = tan A.

State and draw the locus of a swimmer maintaining the same distance from a lighthouse.

State and draw the locus of a point equidistant from two given parallel lines.

Construct a Δ ABC, with AB = 6 cm, AC = BC = 9 cm; find a point 4 cm from A and equidistant from B and C.

Prove that `(cot "A" + "cosec A" - 1)/(cot "A" - "cosec A" + 1) = (1 + cos "A")/sin "A"`

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

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

Prove that `sqrt((1 + cos A)/(1 - cos A)) = (tan A + sin A)/(tan A. sin A)`

Prove that:

`(sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = 2/(2 sin^2 A - 1)`

Using a ruler and compass only: 
(i) Construct a triangle ABC with BC = 6 cm, ∠ABC = 120° and AB = 3.5 cm.
(ii) In the above figure, draw a circle with BC as diameter. Find a point 'P' on the circumference of the circle which is equidistant from Ab and BC.
Measure ∠BCP.

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`.