Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`
Advertisements
उत्तर
We have to prove `(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`
We know that `sin^2 A = cos^2 A = 1`
`So,
`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 = sec A))`
`= (cos^2 A/sin^2 A(1/cos A - 1))/(1 + sin A)`
`= (cos^2 A/sin^2 A (1 - cos A)/(cos A))/(1 + sin A)`
`= (cos A(1 - cos A))/(sin^2 A(1 + sin A))`
`= (cos A (1 - cos A))/((1 - cos^2 A)(1 + sin A))`
`= (cos A (1 - cos A))/((1 - cos A)(1 + cos A)(1 + sin A))`
`= cos A/((1 + cos A)(1 + sin A))`
`= (1/sec A)/((1 + 1/sec A)(1 + sin A))`
`= (1/sec A)/(((sec A + 1)/sec A)) (1 + sin A)`
`= 1/((sec A +1)(1 + sin A))`
Multiplying both the numerator and denominator by (1 - sin A), we have
`= (1 - sin A)/((sec A + 1)(1 + sin A)(1 - sin A))`
`= (1 - sin A)/((sec A + 1)(1 - sin^2 A))`
`= (1 - sin A)/((sec A + 1)cos^2 A)`
`= sec^2 A ((1 - sin A))/((sec A + 1))`
`= sec^2 A ((1 - sin A)/(1 + sec A))`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities
(1 + cot2 A) sin2 A = 1
Prove the following trigonometric identities.
tan2θ cos2θ = 1 − cos2θ
Prove the following trigonometric identities.
`(1 + cos A)/sin^2 A = 1/(1 - cos A)`
Prove the following identities:
(cosec A – sin A) (sec A – cos A) (tan A + cot A) = 1
Prove the following identities:
`(1 + sinA)/cosA + cosA/(1 + sinA) = 2secA`
Show that : `sinA/sin(90^circ - A) + cosA/cos(90^circ - A) = sec A cosec A`
Prove the following identities:
`(1 - cosA)/sinA + sinA/(1 - cosA)= 2cosecA`
Prove the following identities:
(1 + tan A + sec A) (1 + cot A – cosec A) = 2
Show that none of the following is an identity:
(i) `cos^2theta + cos theta =1`
If tan A = n tan B and sin A = m sin B , prove that `cos^2 A = ((m^2-1))/((n^2 - 1))`
2 (sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) is equal to
(sec A + tan A) (1 − sin A) = ______.
Prove the following identity :
`((1 + tan^2A)cotA)/(cosec^2A) = tanA`
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`
Prove that ( 1 + tan A)2 + (1 - tan A)2 = 2 sec2A
Prove that `(sin 70°)/(cos 20°) + (cosec 20°)/(sec 70°) - 2 cos 70° xx cosec 20°` = 0.
Prove the following identities.
`(1 - tan^2theta)/(cot^2 theta - 1)` = tan2 θ
Prove that `sqrt((1 + cos "A")/(1 - cos"A"))` = cosec A + cot A
Prove that sin6A + cos6A = 1 – 3sin2A . cos2A
Statement 1: sin2θ + cos2θ = 1
Statement 2: cosec2θ + cot2θ = 1
Which of the following is valid?
