English Medium इयत्ता १० - CBSE Question Bank Solutions for Mathematics

If m=(acosθ + bsinθ) and n=(asinθ – bcosθ) prove that m²+n²=a²+b²

If acosθ – bsinθ = c, prove that asinθ + bcosθ = `\pm \sqrt{a^{2}+b^{2}-c^{2}`

Prove that: `(1 – sinθ + cosθ)^2 = 2(1 + cosθ)(1 – sinθ)`

If sinθ + sin² θ = 1, prove that cos² θ + cos⁴ θ = 1

Prove that `\frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }+\frac{\sin\theta +\cos \theta }{\sin \theta -\cos \theta }=\frac{2}{2\sin^{2}\theta -1}`

Express the ratios cos A, tan A and sec A in terms of sin A.

Prove that ` \frac{\sin \theta -\cos \theta +1}{\sin\theta +\cos \theta -1}=\frac{1}{\sec \theta -\tan \theta }` using the identity sec² θ = 1 + tan² θ.

Evaluate

`(sin ^2 63^@ + sin^2 27^@)/(cos^2 17^@+cos^2 73^@)`

 Evaluate sin25° cos65° + cos25° sin65°

9 sec² A − 9 tan² A = ______.

(1 + tan θ + sec θ) (1 + cot θ − cosec θ) = ______.

(secA + tanA) (1 − sinA) = ______.

`(1+tan^2A)/(1+cot^2A)` = ______.

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(cosec  θ  – cot θ)^2 = (1-cos theta)/(1 + cos theta)`

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`cos A/(1 + sin A) + (1 + sin A)/cos A = 2 sec A`

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(tan theta)/(1-cot theta) + (cot theta)/(1-tan theta) = 1+secthetacosectheta`

[Hint: Write the expression in terms of sinθ and cosθ]

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(1+ secA)/sec A = (sin^2A)/(1-cosA)` 

[Hint : Simplify LHS and RHS separately.]

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`sqrt((1+sinA)/(1-sinA)) = secA + tanA`

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

`(sintheta - 2sin^3theta)/(2costheta - costheta) =tan theta`

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(cos A-sinA+1)/(cosA+sinA-1)=cosecA+cotA ` using the identity cosec² A = 1 cot² A.