Advertisements
Advertisements
Question
Prove the following trigonometric identities
sec4 A(1 − sin4 A) − 2 tan2 A = 1
Advertisements
Solution
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
RELATED QUESTIONS
Prove the following trigonometric identities.
`cos theta/(1 + sin theta) = (1 - sin theta)/cos theta`
Prove the following trigonometric identities.
sec A (1 − sin A) (sec A + tan A) = 1
Prove the following trigonometric identities.
`(cos theta)/(cosec theta + 1) + (cos theta)/(cosec theta - 1) = 2 tan theta`
Prove the following identities:
`(1 - sinA)/(1 + sinA) = (secA - tanA)^2`
Prove the following identities:
`(1 - cosA)/sinA + sinA/(1 - cosA)= 2cosecA`
` (sin theta + cos theta )/(sin theta - cos theta ) + ( sin theta - cos theta )/( sin theta + cos theta) = 2/ ((1- 2 cos^2 theta))`
`(1+ tan theta + cot theta )(sintheta - cos theta) = ((sec theta)/ (cosec^2 theta)-( cosec theta)/(sec^2 theta))`
Write the value of `4 tan^2 theta - 4/ cos^2 theta`
If 5 `tan theta = 4,"write the value of" ((cos theta - sintheta))/(( cos theta + sin theta))`
Write the value of cos1° cos 2°........cos180° .
Write True' or False' and justify your answer the following :
The value of \[\sin \theta\] is \[x + \frac{1}{x}\] where 'x' is a positive real number .
\[\frac{\tan \theta}{\sec \theta - 1} + \frac{\tan \theta}{\sec \theta + 1}\] is equal to
If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then
9 sec2 A − 9 tan2 A is equal to
Prove the following identity :
`sinθ(1 + tanθ) + cosθ(1 +cotθ) = secθ + cosecθ`
Evaluate:
sin2 34° + sin2 56° + 2 tan 18° tan 72° – cot2 30°
Verify that the points A(–2, 2), B(2, 2) and C(2, 7) are the vertices of a right-angled triangle.
Prove that `sqrt((1 + sin θ)/(1 - sin θ))` = sec θ + tan θ.
Prove that `tan A/(1 + tan^2 A)^2 + cot A/(1 + cot^2 A)^2 = sin A.cos A`
Prove that `sqrt((1 + cos A)/(1 - cos A)) = "cosec" A + cot A`.
