Advertisements
Advertisements
Question
Prove the following trigonometric identities. `(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`
Advertisements
Solution 1
We need to prove `(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`
Now, rationalising the L.H.S, we get
`(1 - cos A)/(1 + cos A) = ((1 - cos A)/(1 + cos A)) ((1 - cos A)/(1 - cos A))`
`= (1 - cos A)^2/(1 - cos^2 A)` (using `a^2 - b^2 = (a + b)(a - b))`
` = (1 + cos^2 A - 2 cos A)/sin^2 A` (Using `sin^2 theta = 1 - cos^2 theta`)
`= 1/sin^2 A + cos^2 A/sin^2 A - (2 cos A)/sin^2 A`
Using `cosec theta = 1/sin theta` and `cot theta = cos theta/sin theta` we get
`1/sin^2 A + cos^2 A/sin^2 A - (2 cos A)/sin^2 A = cosec^2 A + cot^2 A - 2 cot A cosec A`
` (cot A - cosec A)^2` (Using `(a + b)^2 = a^2 + b^2 + 2ab`)
Hence proved.
Solution 2
LHS = `(1 - cos θ)/(1 + cos θ)`
= `(1 - cos θ)/(1 + cos θ) xx (1 - cos θ)/(1 - cos θ)`
= `(1 - cos θ)^2/(1 - cos^2 θ)`
= `(1 - cos θ)^2/(sin^2 θ)`
= `[(1 - cosθ)/(sin θ)]^2`
= `[ 1/sinθ - cosθ/sin θ ]^2`
= ( cosec θ - cot θ )2
= [ - (cot θ - cosec θ)]2
= (cot θ - cosec θ)2
= RHS
Hence proved.
APPEARS IN
RELATED QUESTIONS
`"If "\frac{\cos \alpha }{\cos \beta }=m\text{ and }\frac{\cos \alpha }{\sin \beta }=n " show that " (m^2 + n^2 ) cos^2 β = n^2`
Prove that ` \frac{\sin \theta -\cos \theta +1}{\sin\theta +\cos \theta -1}=\frac{1}{\sec \theta -\tan \theta }` using the identity sec2 θ = 1 + tan2 θ.
Prove the following trigonometric identities.
`tan theta - cot theta = (2 sin^2 theta - 1)/(sin theta cos theta)`
Prove that:
`(cos^3A + sin^3A)/(cosA + sinA) + (cos^3A - sin^3A)/(cosA - sinA) = 2`
If x = a cos θ and y = b cot θ, show that:
`a^2/x^2 - b^2/y^2 = 1`
Prove that:
`cot^2A/(cosecA - 1) - 1 = cosecA`
`(cos theta cosec theta - sin theta sec theta )/(costheta + sin theta) = cosec theta - sec theta`
Prove that `( sintheta - 2 sin ^3 theta ) = ( 2 cos ^3 theta - cos theta) tan theta`
What is the value of \[6 \tan^2 \theta - \frac{6}{\cos^2 \theta}\]
If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, then\[\frac{x^2}{a^2} + \frac{y^2}{b^2}\]
Prove the following identity :
( 1 + cotθ - cosecθ) ( 1 + tanθ + secθ)
Prove the following identity :
`(1 - tanA)^2 + (1 + tanA)^2 = 2sec^2A`
If `x/(a cosθ) = y/(b sinθ) "and" (ax)/cosθ - (by)/sinθ = a^2 - b^2 , "prove that" x^2/a^2 + y^2/b^2 = 1`
Without using trigonometric table , evaluate :
`sin72^circ/cos18^circ - sec32^circ/(cosec58^circ)`
Prove that:
tan (55° + x) = cot (35° – x)
Express (sin 67° + cos 75°) in terms of trigonometric ratios of the angle between 0° and 45°.
Prove that sin2 θ + cos4 θ = cos2 θ + sin4 θ.
Without using a trigonometric table, prove that
`(cos 70°)/(sin 20°) + (cos 59°)/(sin 31°) - 8sin^2 30° = 0`.
Prove the following identities.
tan4 θ + tan2 θ = sec4 θ – sec2 θ
sec2θ – tan2θ = ?
Prove that `"cosec" θ xx sqrt(1 - cos^2θ) = 1`.
Prove that sec2θ – cos2θ = tan2θ + sin2θ.
If 2sin2β − cos2β = 2, then β is ______.
If 1 + sin2α = 3 sinα cosα, then values of cot α are ______.
sin(45° + θ) – cos(45° – θ) is equal to ______.
(tan θ + 2)(2 tan θ + 1) = 5 tan θ + sec2θ.
If sin θ + cos θ = p and sec θ + cosec θ = q, then prove that q(p2 – 1) = 2p.
Show that: `tan "A"/(1 + tan^2 "A")^2 + cot "A"/(1 + cot^2 "A")^2 = sin"A" xx cos"A"`
Prove the following that:
`tan^3θ/(1 + tan^2θ) + cot^3θ/(1 + cot^2θ)` = secθ cosecθ – 2 sinθ cosθ
Prove that `(1 + tan^2 A)/(1 + cot^2 A)` = sec2 A – 1
