Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
(sec A − cosec A) (1 + tan A + cot A) = tan A sec A − cot A cosec A
Advertisements
उत्तर
We have to prove (sec A − cosec A) (1 + tan A + cot A) = tan A sec A − cot A cosec A
We know that `sin^2 A + cos^2 A = 1`
So,
`(sec A − cosec A) (1 + tan A + cot A) = (1/cos A - 1/sin A)(1 + sinA/cos A + cos A/sin A)`
`= ((sin A - cos A)/(sin A cos A))((sin A cos A + sin^2 A + cos^2 A)/(sin A cos A))`
`= ((sin A - cos A)/(sin A cos A)) ((sin A cos A + 1)/(sin A cos A))`
`= ((sin A - cos A)(sin A cos A + 1))/(sin^2 A cos^2 A)`
`= (sin^2 A cos A + sin A - cos^2 A sin A - cos A)/(sin^2 A cos^2 A)`
`= ((sin^2 A cos A - cos A) + (sin A - cos^2 A sin A))/(sin^2 A cos^2 A)`
`= (cos A(sin^2 A - 1) + sin A (1 - cos^2 A))/(sin^2 A cos^2 A)`
`= (cos A(-cos^2 A) + sin A (sin^2 A))/(sin^2 A cos^2 A)`
`= (-cos^3 A + sin^3 A)/(sin^2 A cos^2 A)`
`= (sin^3 A - cos^3 A)/(sin^2 A cos^2 A)`
`= sin^3 A/(sin^2 A cos^2 A) - cos^3 A/(sin^2 A cos^2 A)`
`= sin A/cos^2 A = cos A/sin^2 A`
`= sin A/cos A 1/cos A - cos A/sin A 1/sin A`
= tan A sec A - cot A cosec A
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities:
`(\text{i})\text{ }\frac{\sin \theta }{1-\cos \theta }=\text{cosec}\theta+\cot \theta `
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 trigonometric identities.
`(sec A - tan A)^2 = (1 - sin A)/(1 + sin A)`
Prove the following identities:
(cos A + sin A)2 + (cos A – sin A)2 = 2
Prove the following identities:
`1/(sinA + cosA) + 1/(sinA - cosA) = (2sinA)/(1 - 2cos^2A)`
Write the value of `3 cot^2 theta - 3 cosec^2 theta.`
What is the value of (1 − cos2 θ) cosec2 θ?
Prove the following identity :
`(cosecθ)/(tanθ + cotθ) = cosθ`
Verify that the points A(–2, 2), B(2, 2) and C(2, 7) are the vertices of a right-angled triangle.
Prove that `sqrt((1 - sin θ)/(1 + sin θ)) = sec θ - tan θ`.
Prove that : `(sin(90° - θ) tan(90° - θ) sec (90° - θ))/(cosec θ. cos θ. cot θ) = 1`
Prove that `cos θ/sin(90° - θ) + sin θ/cos (90° - θ) = 2`.
If x = h + a cos θ, y = k + b sin θ.
Prove that `((x - h)/a)^2 + ((y - k)/b)^2 = 1`.
Prove the following identities.
`(sin^3"A" + cos^3"A")/(sin"A" + cos"A") + (sin^3"A" - cos^3"A")/(sin"A" - cos"A")` = 2
The value of sin2θ + `1/(1 + tan^2 theta)` is equal to
`5/(sin^2theta) - 5cot^2theta`, complete the activity given below.
Activity:
`5/(sin^2theta) - 5cot^2theta`
= `square (1/(sin^2theta) - cot^2theta)`
= `5(square - cot^2theta) ......[1/(sin^2theta) = square]`
= 5(1)
= `square`
Prove that `(sintheta + tantheta)/cos theta` = tan θ(1 + sec θ)
Prove that sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ
If 5 tan β = 4, then `(5 sin β - 2 cos β)/(5 sin β + 2 cos β)` = ______.
Let α, β be such that π < α – β < 3π. If sin α + sin β = `-21/65` and cos α + cos β = `-27/65`, then the value of `cos (α - β)/2` is ______.
