Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`1/(sec A + tan A) - 1/cos A = 1/cos A - 1/(sec A - tan A)`
Prove the following:
`1/(sec A + tan A) - 1/cos A = 1/cos A - 1/(sec A - tan A)`
Advertisements
उत्तर
In the given question, we need to prove `1/(sec A + tan A) - 1/cos A = 1/cos A - 1/(sec A - tan A)`
Here, we will first solve the L.H.S.
Now using `sec theta = 1/cos theta` and `tan theta = sin theta/cos theta`, we get
`1/(sec A + tan A) - 1/cos A = 1/(1/cos A + sin A/cos A) - (1/cos A)`
`= 1/(((1 + sin A)/cos A)) - (1/cos A)`
`= (cos A/(1 + sin A)) - (1/cos A)`
`= (cos^2 A - (1 + sin A))/((1 + sin A)(cos A))`
On further solving, we get
`(cos^2 A -(1 + sin A))/((1 + sin A)(cos A)) = (cos^2 A - 1 - sin A)/((1 + sin A)(cos A))`
`= (-sin^2 A - sin A)/((1 + sin A)(cos A))` (Using `sin^2 theta = 1 - cos^2 theta)`
`= (-sin A(sin A + 1))/((1 + sin A)(cos A))`
`= (-sin A)/cos A`
= − tan A
Similarly, we solve the R.H.S.
`((1 - sin A) - cos^2 A)/((cos A)(1 - sin^2 A)) = (1 - sin A - cos^2 A)/((cos A)(1 - sin A))`
`= (sin^2 A - sin A)/((cos A)(1 - sin A))` (Using `sin^2 theta = 1- cos^2 theta`)
`= (-sin A(1 - sin A))/((cos A)(1 - sin A))`
`= (-sin A)/cos A`
= − tan A
So, L.H.S = R.H.S
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove the following identities:
`(i) cos4^4 A – cos^2 A = sin^4 A – sin^2 A`
`(ii) cot^4 A – 1 = cosec^4 A – 2cosec^2 A`
`(iii) sin^6 A + cos^6 A = 1 – 3sin^2 A cos^2 A.`
If tanθ + sinθ = m and tanθ – sinθ = n, show that `m^2 – n^2 = 4\sqrt{mn}.`
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 that (cosec A – sin A)(sec A – cos A) sec2 A = tan A.
Prove the following trigonometric identities:
`(1 - cos^2 A) cosec^2 A = 1`
Prove the following trigonometric identities.
tan2θ cos2θ = 1 − cos2θ
Prove the following trigonometric identities.
sin2 A cot2 A + cos2 A tan2 A = 1
Prove the following trigonometric identities.
`(1 + cos A)/sin^2 A = 1/(1 - cos A)`
`Prove the following trigonometric identities.
`(sec A - tan A)^2 = (1 - sin A)/(1 + sin A)`
Prove the following trigonometric identities.
(cosec θ − sec θ) (cot θ − tan θ) = (cosec θ + sec θ) ( sec θ cosec θ − 2)
Prove the following trigonometric identities.
`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`
Prove that `sqrt((1 + cos theta)/(1 - cos theta)) + sqrt((1 - cos theta)/(1 + cos theta)) = 2 cosec theta`
Prove the following identities:
sec2A + cosec2A = sec2A . cosec2A
Prove the following identities:
(cos A + sin A)2 + (cos A – sin A)2 = 2
Prove the following identities:
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
Prove that:
`(cosecA - sinA)(secA - cosA) = 1/(tanA + cotA)`
If m = a sec A + b tan A and n = a tan A + b sec A, then prove that : m2 – n2 = a2 – b2
Prove the following identities:
(1 + tan A + sec A) (1 + cot A – cosec A) = 2
If sec A + tan A = p, show that:
`sin A = (p^2 - 1)/(p^2 + 1)`
`cos^2 theta /((1 tan theta))+ sin ^3 theta/((sin theta - cos theta))=(1+sin theta cos theta)`
Write the value of ` sin^2 theta cos^2 theta (1+ tan^2 theta ) (1+ cot^2 theta).`
If 5 `tan theta = 4,"write the value of" ((cos theta - sintheta))/(( cos theta + sin theta))`
If `cot theta = 1/ sqrt(3) , "write the value of" ((1- cos^2 theta))/((2 -sin^2 theta))`
Prove that:
`"tanθ"/("secθ" – 1) = (tanθ + secθ + 1)/(tanθ + secθ - 1)`
If cosec θ − cot θ = α, write the value of cosec θ + cot α.
If cosec θ = 2x and \[5\left( x^2 - \frac{1}{x^2} \right)\] \[2\left( x^2 - \frac{1}{x^2} \right)\]
\[\frac{1 + \tan^2 A}{1 + \cot^2 A}\]is equal to
Prove that:
(cosec θ - sinθ )(secθ - cosθ ) ( tanθ +cot θ) =1
Prove the following identity :
tanA+cotA=secAcosecA
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
Prove the following identity :
`(cotA + tanB)/(cotB + tanA) = cotAtanB`
Prove the following identity :
`1/(cosA + sinA - 1) + 2/(cosA + sinA + 1) = cosecA + secA`
Prove the following identity :
`(cot^2θ(secθ - 1))/((1 + sinθ)) = sec^2θ((1-sinθ)/(1 + secθ))`
If x = r sinA cosB , y = r sinA sinB and z = r cosA , prove that `x^2 + y^2 + z^2 = r^2`
For ΔABC , prove that :
`sin((A + B)/2) = cos"C/2`
Evaluate:
`(tan 65^circ)/(cot 25^circ)`
Prove that:
`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = 2cosectheta`
Prove that: 2(sin6 θ + cos6 θ) – 3 (sin4 θ + cos4 θ) + 1 = 0.
Prove that ( 1 + tan A)2 + (1 - tan A)2 = 2 sec2A
Prove that `(tan θ)/(cot(90° - θ)) + (sec (90° - θ) sin (90° - θ))/(cosθ. cosec θ) = 2`.
Prove that `"cosec" θ xx sqrt(1 - cos^2θ) = 1`.
If `tan θ = 9/40`, complete the activity to find the value of sec θ.
Activity:
sec2θ = 1 + `square` ...[Fundamental trigonometric identity]
sec2θ = 1 + `square^2`
sec2θ = 1 + `square`
sec θ = `square`
Prove that `(sin θ + "cosec" θ)/(sin θ) = 2 + cot^2θ`.
Prove that sin2A . tan A + cos2A . cot A + 2 sin A . cos A = tan A + cot A.
If cos A + cos2A = 1, then sin2A + sin4A = ?
If cosec A – sin A = p and sec A – cos A = q, then prove that `(p^2q)^(2/3) + (pq^2)^(2/3) = 1`.
Prove the following:
`sintheta/(1 + cos theta) + (1 + cos theta)/sintheta` = 2cosecθ
The value of 2sinθ can be `a + 1/a`, where a is a positive number, and a ≠ 1.
If `sqrt(3) tan θ` = 1, then find the value of sin2θ – cos2θ.
Show that `(cos^2(45^circ + θ) + cos^2(45^circ - θ))/(tan(60^circ + θ) tan(30^circ - θ)) = 1`
