Advertisements
Advertisements
प्रश्न
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(cos A-sinA+1)/(cosA+sinA-1)=cosecA+cotA ` using the identity cosec2 A = 1 cot2 A.
Advertisements
उत्तर
`(cos A-sinA+1)/(cosA+sinA-1)=cosecA+cotA`
Using the identity cosec2A = 1 + cot2A,
L.H.S = `(cos A-sinA+1)/(cosA+sinA-1)`
= `(cosA/sinA-sinA/sinA+1/sinA)/(cosA/sinA+sinA/sinA+1/sinA)`
= `(cotA-1+cosec A)/(cotA+1-cosec A)`
= `({(cotA)-(1-cosec A)}{(cotA)-(1-cosec A)})/({(cotA)+(1-cosec A)}{(cotA)-(1-cosec A)})`
= `(cot A - 1 + cosecA)^2/((cotA)^2-(1-cosecA)^2)`
= `(cot^2A+1+cosec^2A-2cotA-2cosec A+2cotAcosec A)/(cot^2A-(1+cosec^2 A-2cosec A))`
= `(2cosec^2 A+2cotAcosec A-2cotA-2cosec A)/(cot^2A-1-1cosec^2 A+2cosec A)`
= `(2cosec A(cosecA+cotA)-2(cotA+cosec A))/(cot^2A-cosec^2A-1+2cosec A)`
= `((cosec A+cotA)(2cosec A-2))/(-1-1+2cosec A)`
= `((cosec A+cotA)(2cosec A-2))/(2cosec A-2)`
= cosec A + cot A
= R.H.S
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities:
`(1 - cos theta)/sin theta = sin theta/(1 + cos theta)`
Prove the following trigonometric identities.
`1/(sec A - 1) + 1/(sec A + 1) = 2 cosec A cot A`
Prove the following trigonometric identity.
`(sin theta - cos theta + 1)/(sin theta + cos theta - 1) = 1/(sec theta - tan theta)`
Prove the following trigonometric identities.
(1 + cot A − cosec A) (1 + tan A + sec A) = 2
If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, show that `x^2/a^2 + y^2/b^2 - z^2/c^2 = 1`
Prove the following identities:
`sinA/(1 - cosA) - cotA = cosecA`
` tan^2 theta - 1/( cos^2 theta )=-1`
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
If m = a secA + b tanA and n = a tanA + b secA , prove that m2 - n2 = a2 - b2
Without using trigonometric identity , show that :
`cos^2 25^circ + cos^2 65^circ = 1`
Prove that sin4θ - cos4θ = sin2θ - cos2θ
= 2sin2θ - 1
= 1 - 2 cos2θ
If A = 60°, B = 30° verify that tan( A - B) = `(tan A - tan B)/(1 + tan A. tan B)`.
Without using the trigonometric table, prove that
tan 10° tan 15° tan 75° tan 80° = 1
Prove the following identities.
tan4 θ + tan2 θ = sec4 θ – sec2 θ
Prove the following identities.
`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ
If x sin3 θ + y cos3 θ = sin θ cos θ and x sin θ = y cos θ, then prove that x2 + y2 = 1
cot θ . tan θ = ?
`(1 - tan^2 45^circ)/(1 + tan^2 45^circ)` = ?
If 2sin2θ – cos2θ = 2, then find the value of θ.
Show that: `tan "A"/(1 + tan^2 "A")^2 + cot "A"/(1 + cot^2 "A")^2 = sin"A" xx cos"A"`
