Advertisements
Advertisements
Question
Prove that:
sec2θ + cosec2θ = sec2θ x cosec2θ
Advertisements
Solution
L.H.S = sec2θ + cosec2θ
= 1 + tan2θ + 1 + cot2θ .....[∵ sec2θ = 1 + tan2θ and cosec2θ = 1 + cot2θ]
= 2 + tan2θ + cot2θ .....(i)
R.H.S = sec2θ x cosec2θ
= (1 + tan2θ) x (1 + cot2θ) .....[∵ sec2θ = 1 + tan2θ and cosec2θ = 1 + cot2θ]
= 1 + cot2θ + tan2θ + tan2θ x cot2θ
= 1 + cot2θ + tan2θ + tan2θ x (1/tan2θ) ...... [∵ cot2θ = 1/tan2θ]
= 2 + tan2θ + cot2θ .......(ii)
From (i) and (ii)
sec2θ + cosec2θ = sec2θ x cosec2θ
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities.
`cos theta/(1 + sin theta) = (1 - sin theta)/cos theta`
`Prove the following trigonometric identities.
`(sec A - tan A)^2 = (1 - sin A)/(1 + sin A)`
Prove the following identities:
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
Prove the following identities:
`(sinAtanA)/(1 - cosA) = 1 + secA`
Prove the following identities:
(1 + cot A – cosec A)(1 + tan A + sec A) = 2
If x cos A + y sin A = m and x sin A – y cos A = n, then prove that : x2 + y2 = m2 + n2
If `cosA/cosB = m` and `cosA/sinB = n`, show that : (m2 + n2) cos2 B = n2.
Prove the following identities:
`(1 - 2sin^2A)^2/(cos^4A - sin^4A) = 2cos^2A - 1`
Prove that:
`1/(sinA - cosA) - 1/(sinA + cosA) = (2cosA)/(2sin^2A - 1)`
`(sec^2 theta -1)(cosec^2 theta - 1)=1`
Show that none of the following is an identity:
`sin^2 theta + sin theta = 2`
If `(x/a sin theta - y/b cos theta) = 1` and `(x/a cos theta + y/b sin theta) = 1`, prove that `(x^2/a^2 + y^2/b^2) = 2`.
Write the value of `cosec^2 theta (1+ cos theta ) (1- cos theta).`
Simplify
sin A `[[sinA -cosA],["cos A" " sinA"]] + cos A[[ cos A" sin A " ],[-sin A" cos A"]]`
Prove the following identity :
`cos^4A - sin^4A = 2cos^2A - 1`
Prove that: sin6θ + cos6θ = 1 - 3sin2θ cos2θ.
Prove the following identities.
`costheta/(1 + sintheta)` = sec θ – tan θ
If `(cos alpha)/(cos beta)` = m and `(cos alpha)/(sin beta)` = n, then prove that (m2 + n2) cos2 β = n2
If 3 sin θ = 4 cos θ, then sec θ = ?
If tan θ = 3, then `(4 sin theta - cos theta)/(4 sin theta + cos theta)` is equal to ______.
