Advertisements
Advertisements
Question
Prove that: `sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta`
Advertisements
Solution
LHS = `sqrt((1/cos theta - 1)/(1/cos theta + 1)) + sqrt((1/cos theta +1)/(1/cos theta - 1))`
`= sqrt(((1 - cos theta)/cos theta)/((1+ cos theta)/cos theta)) + sqrt(((1 + cos theta)/cos theta)/((1 - cos theta)/cos theta)`
`= sqrt((1 - cos theta)/(1 + cos theta)) +sqrt((1 + cos theta)/(1 - cos theta))`
`= sqrt((1 - cos theta)/(1 + cos theta) xx (1 - cos theta)/(1 - cos theta)) + sqrt((1 + cos theta)/(1 - cos theta) xx (1 + cos theta)/(1 + cos theta))`
`= sqrt((1 - cos theta)^2/(1 - cos^2 theta)) + sqrt((1 + cos theta)^2/(1 - cos^2 theta))`
`=(1 - cos theta)/sin theta + (1 + cos theta)/sin theta`
`= (1 - cos theta + 1 + cos theta)/sin theta`
`= 2/sin theta`
= 2 cosec
APPEARS IN
RELATED QUESTIONS
if `cosec theta - sin theta = a^3`, `sec theta - cos theta = b^3` prove that `a^2 b^2 (a^2 + b^2) = 1`
Prove the following trigonometric identities.
if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`
Prove that:
(sec A − tan A)2 (1 + sin A) = (1 − sin A)
Prove that:
(1 + tan A . tan B)2 + (tan A – tan B)2 = sec2 A sec2 B
Show that : `sinAcosA - (sinAcos(90^circ - A)cosA)/sec(90^circ - A) - (cosAsin(90^circ - A)sinA)/(cosec(90^circ - A)) = 0`
`(sec theta + tan theta )/( sec theta - tan theta ) = ( sec theta + tan theta )^2 = 1+2 tan^2 theta + 25 sec theta tan theta `
Show that none of the following is an identity:
(i) `cos^2theta + cos theta =1`
Write the value of `(sin^2 theta 1/(1+tan^2 theta))`.
Write the value of `3 cot^2 theta - 3 cosec^2 theta.`
Prove that:
`"tanθ"/("secθ" – 1) = (tanθ + secθ + 1)/(tanθ + secθ - 1)`
\[\frac{\sin \theta}{1 + \cos \theta}\]is equal to
Prove the following identity :
`(1 - cos^2θ)sec^2θ = tan^2θ`
Prove the following identity :
secA(1 + sinA)(secA - tanA) = 1
Prove that cosec2 (90° - θ) + cot2 (90° - θ) = 1 + 2 tan2 θ.
Prove that `sin^2 θ/ cos^2 θ + cos^2 θ/sin^2 θ = 1/(sin^2 θ. cos^2 θ) - 2`.
Prove the following identities: sec2 θ + cosec2 θ = sec2 θ cosec2 θ.
Prove the following identities:
`(1 - tan^2 θ)/(cot^2 θ - 1) = tan^2 θ`.
Prove that: `(sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = 2/(sin^2 A - cos^2 A)`.
If `cos theta/(1 + sin theta) = 1/"a"`, then prove that `("a"^2 - 1)/("a"^2 + 1)` = sin θ
If 2 cos θ + sin θ = `1(θ ≠ π/2)`, then 7 cos θ + 6 sin θ is equal to ______.
