Advertisements
Advertisements
Question
If tan θ – sin2θ = cos2θ, then show that `sin^2θ = 1/2`.
Advertisements
Solution
tan θ – sin2θ = cos2θ ...[Given]
∴ tan θ = sin2θ + cos2θ
∴ tan θ = 1 ...[∵ sin2θ + cos2θ = 1]
But, tan 45° = 1
∴ tan θ = tan 45°
∴ θ = 45°
sin2θ = sin245°
= `(1/sqrt(2))^2`
= `1/2`
APPEARS IN
RELATED QUESTIONS
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 trigonometric identities.
`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`
Prove the following trigonometric identities.
`(cot A + tan B)/(cot B + tan A) = cot A tan B`
Prove that:
`(cos^3A + sin^3A)/(cosA + sinA) + (cos^3A - sin^3A)/(cosA - sinA) = 2`
If sin A + cos A = m and sec A + cosec A = n, show that : n (m2 – 1) = 2 m
Prove the following identities:
`1/(cosA + sinA) + 1/(cosA - sinA) = (2cosA)/(2cos^2A - 1)`
`sqrt((1 + sin θ)/(1 - sin θ)) = sec θ + tan θ`
`(1+ cos theta + sin theta)/( 1+ cos theta - sin theta )= (1+ sin theta )/(cos theta)`
Write the value of `(1+ tan^2 theta ) ( 1+ sin theta ) ( 1- sin theta)`
If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then
Without using trigonometric table , evaluate :
`cos90^circ + sin30^circ tan45^circ cos^2 45^circ`
Prove that (sin θ + cosec θ)2 + (cos θ + sec θ)2 = 7 + tan2 θ + cot2 θ.
`(sin A)/(1 + cos A) + (1 + cos A)/(sin A)` = 2 cosec A
Prove that `cot^2 "A" [(sec "A" - 1)/(1 + sin "A")] + sec^2 "A" [(sin"A" - 1)/(1 + sec"A")]` = 0
sec2θ – tan2θ = ?
Prove that cos2θ . (1 + tan2θ) = 1. Complete the activity given below.
Activity:
L.H.S. = `square`
= `cos^2θ xx square` ...`[1 + tan^2θ = square]`
= `(cos θ xx square)^2`
= 12
= 1
= R.H.S.
If 3 sin θ = 4 cos θ, then sec θ = ?
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.
Prove that `sqrt((1 + cos A)/(1 - cos A)) = "cosec" A + cot A`.
(sec2 θ – 1) (cosec2 θ – 1) is equal to ______.
