Advertisements
Advertisements
Question
Prove that (1 – cos2A) . sec2B + tan2B (1 – sin2A) = sin2A + tan2B.
Advertisements
Solution
L.H.S. = (1 – cos2A) . sec2B + tan2B (1 – sin2A)
= `sin^2A * 1/(cos^2B) + (sin^2B)/(cos^2B) (1 - sin^2A)` ...`[(∵ sin^2A + cos^2A = 1),(∴ 1 - cos^2A = sin^2A)]`
= `(sin^2A)/(cos^2B) + (sin^2B)/(cos^2B) - (sin^2A sin^2B)/(cos^2B)`
= `(sin^2A)/(cos^2B) - (sin^2A sin^2B)/(cos^2B) + (sin^2B)/(cos^2B)`
= `(sin^2A)/(cos^2B) (1 - sin^2B) + tan^2B`
= `(sin^2A)/(cos^2B) (cos^2B) + tan^2B`
= sin2A + tan2B
= R.H.S.
∴ (1 – cos2A) . sec2B + tan2B (1 – sin2A) = sin2A + tan2B
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identity.
`cos^2 A + 1/(1 + cot^2 A) = 1`
Prove the following identities:
cot2 A – cos2 A = cos2 A . cot2 A
Prove the following identities:
(cosec A + sin A) (cosec A – sin A) = cot2 A + cos2 A
If x = r sin A cos B, y = r sin A sin B and z = r cos A, then prove that : x2 + y2 + z2 = r2
If 4 cos2 A – 3 = 0, show that: cos 3 A = 4 cos3 A – 3 cos A
`sin theta (1+ tan theta) + cos theta (1+ cot theta) = ( sectheta+ cosec theta)`
If `sin theta = 1/2 , " write the value of" ( 3 cot^2 theta + 3).`
If `tan theta = 1/sqrt(5), "write the value of" (( cosec^2 theta - sec^2 theta))/(( cosec^2 theta - sec^2 theta))`.
If \[\sin \theta = \frac{1}{3}\] then find the value of 9tan2 θ + 9.
Prove the following identity :
`(cosecθ)/(tanθ + cotθ) = cosθ`
Prove that `cos θ/sin(90° - θ) + sin θ/cos (90° - θ) = 2`.
Prove the following identities.
(sin θ + sec θ)2 + (cos θ + cosec θ)2 = 1 + (sec θ + cosec θ)2
cos θ . sec θ = ?
sec 60° = ?
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.
Prove that sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ.
If tan θ = 3, then `(4 sin theta - cos theta)/(4 sin theta + cos theta)` is equal to ______.
Prove the following:
`tanA/(1 + sec A) - tanA/(1 - sec A)` = 2cosec A
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 ______.
If tan θ = `x/y`, then cos θ is equal to ______.
