SSC (English Medium) १० वीं कक्षा - Maharashtra State Board Question Bank Solutions for Geometry Mathematics 2

Prove that cot²θ × sec²θ = cot²θ + 1.

If 3 sin θ = 4 cos θ, then sec θ = ?

Prove that sec²θ – cos²θ = tan²θ + sin²θ.

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

Prove that `(cos^2θ)/(sinθ) + sin θ = "cosec"  θ`.

Prove that `(cosθ)/(1 + sinθ) = (1 - sinθ)/(cosθ)`.

sin⁴A – cos⁴A = 1 – 2cos²A. For proof of this complete the activity given below.

Activity:

L.H.S. = `square`

 = (sin²A + cos²A) `(square)`

= `1 (square)`   ...`[sin^2"A" + square = 1]`

= `square` – cos²A   ...[sin²A = 1 – cos²A]

= `square`

= R.H.S.

tan²θ – sin²θ = tan²θ × sin²θ. For proof of this complete the activity given below.

Activity:

L.H.S. = `square`

= `square (1 - (sin^2θ)/(tan^2θ))`

= `tan^2θ (1 - square/((sin^2θ)/(cos^2θ)))`

= `tan^2θ (1 - (sin^2θ)/1 xx (cos^2θ)/square)`

= `tan^2θ (1 - square)`

= `tan^2θ xx square`   ...[1 – cos²θ = sin²θ]

= R.H.S.

If `tan θ = 7/24`, then to find value of cos θ complete the activity given below.

Activity:

sec²θ = 1 + `square`   ...[Fundamental tri. identity]

sec²θ = 1 + `square^2`

sec²θ = 1 + `square/576`

sec²θ = `square/576`

sec θ = `square` 

cos θ = `square   ...`[cos theta = 1/sectheta]`

To prove cot θ + tan θ = cosec θ × sec θ, complete the activity given below.

Activity:

L.H.S. = `square`

= `square/(sinθ) + (sinθ)/(cosθ)`

= `(cos^2θ + sin^2θ)/square`

= `1/(sinθ.cosθ)`   ...`[cos^2θ + sin^2θ = square]`

= `1/(sinθ) xx 1/square`

= `square`

= R.H.S.

If 5 sec θ – 12 cosec θ = 0, then find values of sin θ, sec θ.

Prove that `(tan(90 - θ) + cot(90 - θ))/("cosec"  θ) = sec θ`.

Prove that cot²θ – tan²θ = cosec²θ – sec²θ.

Prove that `(sin θ)/(sec θ + 1) + (sin θ)/(sec θ - 1) = 2 cot θ`.

Prove that `(sec A)/(tan A + cot A) = sin A`.

Prove that `(sin θ + "cosec"  θ)/(sin θ) = 2 + cot^2θ`.

Prove that `(cot A)/(1 - cot A) + (tan A)/(1 - tan A) = -1`.

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

Prove that sin⁴A – cos⁴A = 1 – 2 cos²A.

Prove that sec²θ – cos²θ = tan²θ + sin²θ.