Advertisements
Advertisements
Question
Prove the following:
`tanA/(1 + sec A) - tanA/(1 - sec A)` = 2cosec A
Advertisements
Solution
L.H.S:
`tanA/(1 + sec A) - tanA/(1 - sec A)`
Taking LCM of the denominators,
= `(tanA(1 - sec A) - tanA(1 + sec A))/((1 + sec A)(1 - sec A))`
Since, (1 + sec A)(1 – sec A) = 1 – sec2A
= `(tan A(1 - secA - 1 - sec A))/(1 - sec^2A)`
= `(tan A(-2 sec A))/(1 - sec^2 A)`
= `(2 tan A *sec A)/(sec^2 A - 1)`
Since,
sec2A – tan2A = 1
sec2A – 1 = tan2A
= `(2 tan A * sec A)/(tan^2 A)`
Since, sec A = `(1/cosA)` and tan A = `(sinA/cosA)`
= `(2secA)/tanA = (2cosA)/(cosA sinA)`
= `2/sinA`
= 2 cosec A ...`(∵ 1/sinA = "cosec" A)`
= R.H.S
Hence proved.
APPEARS IN
RELATED QUESTIONS
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(1+ secA)/sec A = (sin^2A)/(1-cosA)`
[Hint : Simplify LHS and RHS separately.]
Prove the following trigonometric identities.
`(1 + sec theta)/sec theta = (sin^2 theta)/(1 - cos theta)`
Prove the following trigonometric identities.
`(1 + cos A)/sin A = sin A/(1 - cos A)`
Prove the following trigonometric identities.
`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`
Prove the following identities:
`1/(secA + tanA) = secA - tanA`
Prove the following identities:
`(1 + cosA)/(1 - cosA) = tan^2A/(secA - 1)^2`
Prove the following identities:
cosec4 A (1 – cos4 A) – 2 cot2 A = 1
`(1+ tan^2 theta)/(1+ tan^2 theta)= (cos^2 theta - sin^2 theta)`
`(1+ cos theta + sin theta)/( 1+ cos theta - sin theta )= (1+ sin theta )/(cos theta)`
If `( tan theta + sin theta ) = m and ( tan theta - sin theta ) = n " prove that "(m^2-n^2)^2 = 16 mn .`
Write the value of `cosec^2 theta (1+ cos theta ) (1- cos theta).`
If \[\sin \theta = \frac{1}{3}\] then find the value of 2cot2 θ + 2.
Prove the following identity :
`sinθ(1 + tanθ) + cosθ(1 +cotθ) = secθ + cosecθ`
If sinA + cosA = `sqrt(2)` , prove that sinAcosA = `1/2`
Without using trigonometric table , evaluate :
`cosec49°cos41° + (tan31°)/(cot59°)`
Prove that `sinA/sin(90^circ - A) + cosA/cos(90^circ - A) = sec(90^circ - A) cosec(90^circ - A)`
If A + B = 90°, show that sec2 A + sec2 B = sec2 A. sec2 B.
If sin A = `1/2`, then the value of sec A is ______.
Prove that `(cot A - cos A)/(cot A + cos A) = (cos^2 A)/(1 + sin A)^2`
