Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities
sec4 A(1 − sin4 A) − 2 tan2 A = 1
Advertisements
उत्तर
We have to prove sec4 A(1 − sin4 A) − 2 tan2 A = 1
We know that `sin^2 A + cos^2 A = 1`
So,
`sec^4 A (1 - sin^4 A) - 2tan^2 A = 1/cos^4 A (1 - sin^4 A) - 2 sin^2 A/cos^2 A`
`= (1/cos^4 A - sin^4 A/cos^4 A) - 2 (sin^2 A)/(cos^2 A)`
`= ((1 - sin^4 A)/cos^4 A) - 2 (sin^2 A)/cos^2 A`
`= ((1 - sin^2 A)(1 + sin^2 A))/cos^4 A - 2 sin^2 A/cos^2 A`
`= (cos^2 A (1 + sin^2 A))/cos^4 A - 2 sin^2 A/cos^2 A`
`= (1 + sin^2 A - 2 sin^2 A)/cos^2 A`
`= (1 - sin^2 A)/cos^2 A`
`= cos^2 A/cos^2 A`
= 1
Hence proved.
APPEARS IN
संबंधित प्रश्न
Evaluate without using trigonometric tables:
`cos^2 26^@ + cos 64^@ sin 26^@ + (tan 36^@)/(cot 54^@)`
Prove the following trigonometric identity.
`cos^2 A + 1/(1 + cot^2 A) = 1`
Prove the following trigonometric identities.
`cot theta - tan theta = (2 cos^2 theta - 1)/(sin theta cos theta)`
Prove the following trigonometric identities.
`((1 + tan^2 theta)cot theta)/(cosec^2 theta) = tan theta`
Prove the following trigonometric identity.
`(sin theta - cos theta + 1)/(sin theta + cos theta - 1) = 1/(sec theta - tan theta)`
Prove the following trigonometric identities.
tan2 A sec2 B − sec2 A tan2 B = tan2 A − tan2 B
If sec A + tan A = p, show that:
`sin A = (p^2 - 1)/(p^2 + 1)`
`(1-cos^2theta) sec^2 theta = tan^2 theta`
`(1+ cos theta + sin theta)/( 1+ cos theta - sin theta )= (1+ sin theta )/(cos theta)`
Write the value of tan10° tan 20° tan 70° tan 80° .
If cos (\[\alpha + \beta\]= 0 , then sin \[\left( \alpha - \beta \right)\] can be reduced to
Prove the following identity :
`cos^4A - sin^4A = 2cos^2A - 1`
Prove the following identity :
`cosecA + cotA = 1/(cosecA - cotA)`
Prove the following identity :
`(tanθ + 1/cosθ)^2 + (tanθ - 1/cosθ)^2 = 2((1 + sin^2θ)/(1 - sin^2θ))`
Find the value of x , if `cosx = cos60^circ cos30^circ - sin60^circ sin30^circ`
Prove that `sqrt((1 + sin A)/(1 - sin A))` = sec A + tan A.
`(sin A)/(1 + cos A) + (1 + cos A)/(sin A)` = 2 cosec A
If `cos theta/(1 + sin theta) = 1/"a"`, then prove that `("a"^2 - 1)/("a"^2 + 1)` = sin θ
If `sqrt(3) tan θ` = 1, then find the value of sin2θ – cos2θ.
tan θ × `sqrt(1 - sin^2 θ)` is equal to:
