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Question
Prove that the following identities:
Sec A( 1 + sin A)( sec A - tan A) = 1.
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Solution
LHS = sec A(1 + sin A )( sec A - tan A)
= `1/cos A (1 + sin A) (1/cos A - sin A/cos A)`
= `1/cos A (1 + sin A) ((1 - sin A)/cos A)`
= `(1 - sin^2 A)/(cos^2 A) = (cos^2 A)/(cos^2 A)`
= 1
= RHS
Hence proved.
RELATED QUESTIONS
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`sqrt((1+sinA)/(1-sinA)) = secA + tanA`
If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.
Prove the following identities:
`((1 + tan^2A)cotA)/(cosec^2A) = tan A`
Find A if tan 2A = cot (A-24°).
sin4A – cos4A = 1 – 2cos2A. For proof of this complete the activity given below.
Activity:
L.H.S = `square`
= (sin2A + cos2A) `(square)`
= `1 (square)` .....`[sin^2"A" + square = 1]`
= `square` – cos2A .....[sin2A = 1 – cos2A]
= `square`
= R.H.S
Prove that `sqrt((1 + cos "A")/(1 - cos"A"))` = cosec A + cot A
Prove that sin4A – cos4A = 1 – 2cos2A
Prove that sin6A + cos6A = 1 – 3sin2A . cos2A
Which of the following is true for all values of θ (0° ≤ θ ≤ 90°)?
Prove the following identity:
(sin2θ – 1)(tan2θ + 1) + 1 = 0
