Advertisements
Advertisements
Question
Prove the following trigonometric identities.
`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`
Solution
We have prove that
`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`
We know that `sin^2 A + cos^2 A = 1`
So,
(2 + cot A + tan A)(sin A - cos A)
`= (1 + cos A/sin A + sin A/cos A)(sin A - cos A)`
`= ((sin A cos A + cos^2 A + sin^2 A)/(sin A cos A)) (sin A - cos A)`
`= ((sin A cos A + 1)/(sin A cos A))(sin A - cos A)`
`= ((sin A - cos A)(sin A cos A + 1))/(sin A cos A)`
`= (sin^2 A cos A + sin A - cos^2 A sin A - cos A)/(sin A cos A)`
`= ((sin^2 A cos A - cos A) + (sin A - cos^2 A sin A))/(sin A cos A)`
`= (cos A (sin^2 A - 1)+ sin A (1 - sin^2 A))/(sin A cos A)`
`= (cos A (-cos^2 A) + sin A (sin^2 A))/(sin A cos A)`
`= (-cos^3 A + sin^3 A)/(sin A cos A)`
`= (sin^3 A - cos^3 A)/(sin A cos A)`
`= (sin^2 A)/cos A - cos^2 A/sin A`
`= sin A/cos A sin A - cos A/sin A cos A`
`= tan A sin A - cot A cos A`
= sin A tan A - cos A cot A
Now
`sec A/(cosec^2 A) - (cosec A)/sec^2 A = (1/cos A)/(1/sin^2 A) - (1/sin A)/(1/cos^2 A)`
`= sin^2 A/cos A - cos^2 A/sin A`
`= sin A sin A/cos a - cos A cos A/sin A`
= sin A tan A - cos A cot A
Hence proved.
APPEARS IN
RELATED QUESTIONS
Prove the following identities:
`( i)sin^{2}A/cos^{2}A+\cos^{2}A/sin^{2}A=\frac{1}{sin^{2}Acos^{2}A)-2`
`(ii)\frac{cosA}{1tanA}+\sin^{2}A/(sinAcosA)=\sin A\text{}+\cos A`
`( iii)((1+sin\theta )^{2}+(1sin\theta)^{2})/cos^{2}\theta =2( \frac{1+sin^{2}\theta}{1-sin^{2}\theta } )`
If m=(acosθ + bsinθ) and n=(asinθ – bcosθ) prove that m2+n2=a2+b2
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.
Prove the following trigonometric identities.
`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`
Prove the following trigonometric identities.
`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`
Prove the following identities:
`((1 + tan^2A)cotA)/(cosec^2A) = tan A`
Prove that:
(sec A − tan A)2 (1 + sin A) = (1 − sin A)
Prove the following identities:
`(1 - 2sin^2A)^2/(cos^4A - sin^4A) = 2cos^2A - 1`
`cot^2 theta - 1/(sin^2 theta ) = -1`a
Write the value of `(sin^2 theta 1/(1+tan^2 theta))`.
If ` cot A= 4/3 and (A+ B) = 90° ` ,what is the value of tan B?
Write the value of cos1° cos 2°........cos180° .
Prove the following identity :
`(tanθ + secθ - 1)/(tanθ - secθ + 1) = (1 + sinθ)/(cosθ)`
Prove the following identity :
`tan^2A - sin^2A = tan^2A.sin^2A`
Prove that `sqrt(2 + tan^2 θ + cot^2 θ) = tan θ + cot θ`.
Prove that: sin6θ + cos6θ = 1 - 3sin2θ cos2θ.
Prove that:
`(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos^3 θ - sin^3 θ)/(cos θ - sin θ) = 2`
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`
`sqrt((1 - cos^2theta) sec^2 theta) = tan theta`
Show that: `tan "A"/(1 + tan^2 "A")^2 + cot "A"/(1 + cot^2 "A")^2 = sin"A" xx cos"A"`
