English Medium Class 10 - CBSE Question Bank Solutions for Mathematics

Prove the following identities:

`(cos theta  "cosec"  theta - sin theta sec theta )/(cos theta + sin theta) = "cosec"  theta - sec theta`

Prove the following identities:

`(1 + tan theta + cot theta)(sin theta - cos theta) = ((sec theta)/("cosec"^2 theta) - ("cosec"  theta)/(sec^2 theta))`

Prove the following identities:

`(cot^2 theta (sec theta - 1))/((1 + sin theta)) + (sec^2 theta(sin theta - 1))/((1 + sec theta)) = 0`

`{1/((sec^2 theta- cos^2 theta))+ 1/((cosec^2 theta - sin^2 theta))} ( sin^2 theta cos^2 theta) = (1- sin^2 theta cos ^2 theta)/(2+ sin^2 theta cos^2 theta)`

`((sin A-  sin B ))/(( cos A + cos B ))+ (( cos A - cos B ))/(( sinA + sin B ))=0` 

`(tan A + tanB )/(cot A + cot B) = tan A tan B`

Show that none of the following is an identity:

`cos^2theta + cos theta = 1`

Show that none of the following is an identity: 

`sin^2 theta + sin theta = 2`

Show that none of the following is an identity:

`tan^2 theta + sin theta = cos^2 theta`

Prove that `( sintheta - 2 sin ^3 theta ) = ( 2 cos ^3 theta - cos theta) tan theta`

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

If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that (x² – y²) = (a² – b²).

If `(x/a sin theta - y/b cos theta) = 1` and `(x/a cos theta + y/b sin theta) = 1`, prove that `(x^2/a^2 + y^2/b^2) = 2`.

If` (sec theta + tan theta)= m and ( sec theta - tan theta ) = n ,` show that mn =1

If `( cosec theta + cot theta ) =m and ( cosec theta - cot theta ) = n, ` show that mn = 1.

If x = a cos³θ and y = b sin³θ, prove that `(x/a)^(2/3) + (y/b)^(2/3) = 1`.

If (tan θ + sin θ) = m and (tan θ – sin θ) = n, prove that (m² – n²)² = 16 mn.

If (cot θ + tan θ) = m and (sec θ – cos θ) = n, prove that `(m^2 n)^(2//3) - (mn^2)^(2//3) = 1`.

If (cosec θ – sin θ) = a³ and (sec θ – cos θ) = b³, prove that a²b²(a² + b²) = 1.

If`( 2 sin theta + 3 cos theta) =2 , " prove that " (3 sin theta - 2 cos theta) = +- 3.`