English Medium कक्षा १० - CBSE Question Bank Solutions for Mathematics

if `cosec theta - sin theta = a^3`, `sec theta - cos theta = b^3` prove that `a^2 b^2 (a^2 + b^2) = 1`

if `a cos^3 theta + 3a cos theta sin^2 theta = m, a sin^3 theta + 3 a cos^2 theta sin theta = n`Prove that `(m + n)^(2/3) + (m - n)^(2/3)`

Prove the following trigonometric identities.

if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`

If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.

If a cos θ + b sin θ = m and a sin θ – b cos θ = n, prove that a² + b² = m² + n²

If cos θ + cot θ = m and cosec θ – cot θ = n, prove that mn = 1

Prove the following trigonometric identities.

if cos A + cos² A = 1, prove that sin² A + sin⁴ A = 1

Prove that: `sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta`

Prove that

`sqrt((1 + sin θ)/(1 - sin θ)) + sqrt((1 - sin θ)/(1 + sin θ)) = 2 sec θ`

Prove that `sqrt((1 + cos theta)/(1 - cos theta)) + sqrt((1 - cos theta)/(1 + cos theta)) = 2 cosec theta`

Prove that  `(sec theta - 1)/(sec theta + 1) = ((sin theta)/(1 + cos theta))^2` 

If cos θ + cos² θ = 1, prove that sin¹² θ + 3 sin¹⁰ θ + 3 sin⁸ θ + sin⁶ θ + 2 sin⁴ θ + 2 sin² θ − 2 = 1

Given that:
(1 + cos α) (1 + cos β) (1 + cos γ) = (1 − cos α) (1 − cos α) (1 − cos β) (1 − cos γ)

Show that one of the values of each member of this equality is sin α sin β sin γ

If sin θ + cos θ = x, prove that  `sin^6 theta + cos^6 theta = (4- 3(x^2 - 1)^2)/4`

If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, show that `x^2/a^2 + y^2/b^2 - x^2/c^2 = 1`

(i)` (1-cos^2 theta )cosec^2theta = 1`

`(1 + cot^2 theta ) sin^2 theta =1`

`(sec^2 theta-1) cot ^2 theta=1`

`(sec^2 theta -1)(cosec^2 theta - 1)=1`

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