Advertisements
Advertisements
Question
Prove that `(tan(90 - θ) + cot(90 - θ))/("cosec" θ) = sec θ`.
Advertisements
Solution
L.H.S. = `(tan(90 - θ) + cot(90 - θ))/("cosec" θ)`
= `1/("cosec" θ)(cot θ + tan θ)` ...`[(∵ tan(90 - θ) = cot θ),(cot(90 - θ) = tan θ)]`
= sin θ (cot θ + tan θ)
= `sin θ ((cos θ)/(sin θ) + (sin θ)/(cos θ))`
= `sin θ ((cos^2θ + sin^2θ)/(sinθ cosθ))`
= `sin θ (1/(sin θ cos θ))` ...[∵ sin2θ + cos2θ = 1]
= `1/(cos θ)`
= sec θ
= R.H.S.
∴ `(tan(90 - θ) + cot(90 - θ))/("cosec" θ) = sec θ`
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities.
`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`
Prove the following trigonometric identities.
`tan A/(1 + tan^2 A)^2 + cot A/((1 + cot^2 A)) = sin A cos A`
If m = a sec A + b tan A and n = a tan A + b sec A, then prove that : m2 – n2 = a2 – b2
`1/((1+ sin θ)) + 1/((1 - sin θ)) = 2 sec^2 θ`
`1+ (cot^2 theta)/((1+ cosec theta))= cosec theta`
`1 + (tan^2 θ)/((1 + sec θ)) = sec θ`
` (sin theta + cos theta )/(sin theta - cos theta ) + ( sin theta - cos theta )/( sin theta + cos theta) = 2/ ((1- 2 cos^2 theta))`
If`( 2 sin theta + 3 cos theta) =2 , " prove that " (3 sin theta - 2 cos theta) = +- 3.`
If x = a sin θ and y = bcos θ , write the value of`(b^2 x^2 + a^2 y^2)`
If sin2 θ cos2 θ (1 + tan2 θ) (1 + cot2 θ) = λ, then find the value of λ.
Prove the following identity :
(secA - cosA)(secA + cosA) = `sin^2A + tan^2A`
Prove the following identity :
`(1 + cosA)/(1 - cosA) = tan^2A/(secA - 1)^2`
Without using trigonometric table , evaluate :
`cos90^circ + sin30^circ tan45^circ cos^2 45^circ`
Prove that `( tan A + sec A - 1)/(tan A - sec A + 1) = (1 + sin A)/cos A`.
If x sin3θ + y cos3 θ = sin θ cos θ and x sin θ = y cos θ , then show that x2 + y2 = 1.
Proved that cosec2(90° - θ) - tan2 θ = cos2(90° - θ) + cos2 θ.
Prove that:
`(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos^3 θ - sin^3 θ)/(cos θ - sin θ) = 2`
If tan θ × A = sin θ, then A = ?
Prove that `sqrt((1 + cos A)/(1 - cos A)) = "cosec" A + cot A`.
If cos A = `(2sqrt(m))/(m + 1)`, then prove that cosec A = `(m + 1)/(m - 1)`.
