Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta + cot theta`
Advertisements
उत्तर
We have to prove the following identity
`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta + cot theta`
Consider the LHS = `(cos theta - sin theta + 1)/(cos theta + sin theta - 1)`
`= (cos theta - sin theta + 1)/(cos theta + sin theta - 1) xx (cos theta + sin theta + 1)/(cos theta + sin theta + 1)`
`= ((cos theta + 1)^2 - (sin theta)^2)/((cos theta + sin theta)^2 - (1)^2)`
`= (cos^2 theta + 1 + 2 cos theta - sin^2 theta)/(cos^2 theta + sin^2 theta + 2 cos theta sin theta - 1)`
`= (cos^2 theta + 1 + 2 cos theta - (1 - cos^2 theta))/(1 + 2 cos theta sin theta - 1)`
`= (2 cos^2 theta + 2 cos theta)/(2 cos theta sin theta)`
`= (2 cos^2 theta + 2 cos theta)/(2 cos theta sin theta)`
`= (2 cos theta(cos theta + 1))/(2 cos theta sin theta)`
`= (cos theta + 1)/sin theta`
`= cos theta/sin theta + 1/sin theta`
`= cot theta + cosec theta`
= RHS
APPEARS IN
संबंधित प्रश्न
`"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`
if `cos theta = 5/13` where `theta` is an acute angle. Find the value of `sin theta`
Prove the following trigonometric identities.
`cos^2 A + 1/(1 + cot^2 A) = 1`
Prove the following trigonometric identities.
`tan θ/(1 - cot θ) + cot θ/(1 - tan θ) = 1 + tan θ + cot θ`
Prove the following trigonometric identity:
`sqrt((1 + sin A)/(1 - sin A)) = sec A + tan A`
Prove the following trigonometric identities.
`(cot A + tan B)/(cot B + tan A) = cot A tan B`
Prove the following trigonometric identities.
tan2 A sec2 B − sec2 A tan2 B = tan2 A − tan2 B
If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.
Prove the following trigonometric identities.
If cos A + cos2 A = 1, prove that sin2 A + sin4 A = 1
Prove the following identities:
`((1 + tan^2A)cotA)/(cosec^2A) = tan A`
Prove the following identities:
(cos A + sin A)2 + (cos A – sin A)2 = 2
Prove the following identities:
`sqrt((1 + sinA)/(1 - sinA)) = cosA/(1 - sinA)`
Prove that:
`cot^2A/(cosecA - 1) - 1 = cosecA`
`tan theta/(1+ tan^2 theta)^2 + cottheta/(1+ cot^2 theta)^2 = sin theta cos theta`
If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that (x2 – y2) = (a2 – b2).
If`( 2 sin theta + 3 cos theta) =2 , " prove that " (3 sin theta - 2 cos theta) = +- 3.`
If `cos B = 3/5 and (A + B) =- 90° ,`find the value of sin A.
Prove the following identity :
`(1 + sinθ)/(cosecθ - cotθ) - (1 - sinθ)/(cosecθ + cotθ) = 2(1 + cotθ)`
Without using trigonometric identity , show that :
`sin(50^circ + θ) - cos(40^circ - θ) = 0`
If sec θ = `25/7`, find the value of tan θ.
Solution:
1 + tan2 θ = sec2 θ
∴ 1 + tan2 θ = `(25/7)^square`
∴ tan2 θ = `625/49 - square`
= `(625 - 49)/49`
= `square/49`
∴ tan θ = `square/7` ........(by taking square roots)
