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

Prove the following identity : 

`sin^8θ - cos^8θ = (sin^2θ - cos^2θ)(1 - 2sin^2θcos^2θ)`

Prove the following identity :

`tan^2θ/(tan^2θ - 1) + (cosec^2θ)/(sec^2θ - cosec^2θ) = 1/(sin^2θ - cos^2θ)`

Prove the following identity :

`(sec^2θ - sin^2θ)/tan^2θ = cosec^2θ - cos^2θ`

Prove the following identity :

`(cos^3θ + sin^3θ)/(cosθ + sinθ) + (cos^3θ - sin^3θ)/(cosθ - sinθ) = 2`

Prove the following identity :

`(tanθ + sinθ)/(tanθ - sinθ) = (secθ + 1)/(secθ - 1)`

Prove the following identity : 

`[1/((sec^2θ - cos^2θ)) + 1/((cosec^2θ - sin^2θ))](sin^2θcos^2θ) = (1 - sin^2θcos^2θ)/(2 + sin^2θcos^2θ)`

Prove the following identity :

`(cot^2θ(secθ - 1))/((1 + sinθ)) = sec^2θ((1-sinθ)/(1 + secθ))`

If m = a secA + b tanA and n = a tanA + b secA , prove that m² - n² = a² - b²

If `x/(a cosθ) = y/(b sinθ)   "and"  (ax)/cosθ - (by)/sinθ = a^2 - b^2 , "prove that"  x^2/a^2 + y^2/b^2 = 1`

If x = r sinA cosB , y = r sinA sinB and z = r cosA , prove that   `x^2 + y^2 + z^2 = r^2`

If sinA + cosA = m and secA + cosecA = n , prove that n(m² - 1) = 2m

If x = acosθ , y = bcotθ , prove that `a^2/x^2 - b^2/y^2 = 1.`

If secθ + tanθ = m , secθ - tanθ = n , prove that mn = 1

If x = asecθ + btanθ and y = atanθ + bsecθ , prove that `x^2 - y^2 = a^2 - b^2`

If tanA + sinA = m and tanA - sinA = n , prove that (`m^2 - n^2)^2` = 16mn 

If sinA + cosA = `sqrt(2)` , prove that sinAcosA = `1/2`

If `asin^2θ + bcos^2θ = c and p sin^2θ + qcos^2θ = r` , prove that (b - c)(r - p) = (c - a)(q - r)

Without using trigonometric table , evaluate : 

`cosec49°cos41° + (tan31°)/(cot59°)`

Without using trigonometric table , evaluate : 

`(sin47^circ/cos43^circ)^2 - 4cos^2 45^circ + (cos43^circ/sin47^circ)^2`

Without using trigonometric table , evaluate : 

`cos90^circ + sin30^circ tan45^circ cos^2 45^circ`