Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
(1 + cot A − cosec A) (1 + tan A + sec A) = 2
Advertisements
उत्तर
We have to prove (1 + cot A − cosec A) (1 + tan A + sec A) = 2
We know that, `sin^2 A + cos^2 A = 1`
So.
`(1 + cot A − cosec A) (1 + tan A + sec A) = (1 + cosA/sin A - 1/ sinA) (1 + sin A/cos A + 1/cos A)`
`= ((sin A + cos A - 1)/sin A)((cos A + sin A + 1)/cos A)`
`= ((sin A + cos A -1)(sin A + cos A + 1))/(sin A cos A)`
`= ({(sin A + cos A) - 1}{(sin A + cos A) + 1})/(sin A cos A)`
`= ((sin A + cos A)^2 -1)/(sin A cos A)`
`= (sin^2 A + 2 sin A cos A + cos^2 A - 1)/(sin A cos A)`
`= ((sin^2 A + cos^2 A) + 2 sin A cos A - 1)/(sin A cos A)`
`= (1 + 2 sin A cos A -1)/(sin A cos A)`
`= (2 sin A cos A)/(sin A cos A)`
= 2
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities.
sec A (1 − sin A) (sec A + tan A) = 1
Prove the following trigonometric identities.
`(1 + cos A)/sin^2 A = 1/(1 - cos A)`
Prove that: `sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta`
Prove the following identities:
`1/(tan A + cot A) = cos A sin A`
Prove the following identities:
`1 - cos^2A/(1 + sinA) = sinA`
Prove that:
(sec A − tan A)2 (1 + sin A) = (1 − sin A)
`sin^2 theta + 1/((1+tan^2 theta))=1`
`(tan^2theta)/((1+ tan^2 theta))+ cot^2 theta/((1+ cot^2 theta))=1`
If `sec theta + tan theta = p,` prove that
(i)`sec theta = 1/2 ( p+1/p) (ii) tan theta = 1/2 ( p- 1/p) (iii) sin theta = (p^2 -1)/(p^2+1)`
If `cos theta = 2/3 , "write the value of" ((sec theta -1))/((sec theta +1))`
Prove the following identity :
`cos^4A - sin^4A = 2cos^2A - 1`
Prove the following identity :
`sin^4A + cos^4A = 1 - 2sin^2Acos^2A`
Prove the following identity :
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (cosA + 1)/sinA`
Prove that `( 1 + sin θ)/(1 - sin θ) = 1 + 2 tan θ/cos θ + 2 tan^2 θ` .
Prove that `sin^2 θ/ cos^2 θ + cos^2 θ/sin^2 θ = 1/(sin^2 θ. cos^2 θ) - 2`.
Prove the following identities: cot θ - tan θ = `(2 cos^2 θ - 1)/(sin θ cos θ)`.
tan (90 – θ) = ?
Prove that sin2A . tan A + cos2A . cot A + 2 sin A . cos A = tan A + cot A.
Prove that sin θ (1 – tan θ) – cos θ (1 – cot θ) = cosec θ – sec θ.
Simplify (1 + tan2θ)(1 – sinθ)(1 + sinθ)
