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If 5 `tan theta = 4,"write the value of" ((cos theta - sintheta))/(( cos theta + sin theta))`
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We have ,
5 `tan theta = 4`
⇒ `tan theta = 4/5`
Now ,
`((cos theta - sintheta))/(( cos theta + sin theta))`
`=(((cos theta )/(cos theta)- (sin theta )/(cos theta)))/((cos theta/ cos theta+ sin theta/ cos theta)` (ЁЭР╖ЁЭСЦЁЭСгЁЭСЦЁЭССЁЭСЦЁЭСЫЁЭСФ ЁЭСЫЁЭСвЁЭСЪЁЭСТЁЭСЯЁЭСОЁЭСбЁЭСЬЁЭСЯ ЁЭСОЁЭСЫЁЭСС ЁЭССЁЭСТЁЭСЫЁЭСЬЁЭСЪЁЭСЦЁЭСЫЁЭСОЁЭСбЁЭСЬЁЭСЯ ЁЭСПЁЭСж cos θ)
`=((1- tan theta))/((1+ tan theta))`
`= ((1/1-4/5))/((1/1+4/5))`
`= ((1/5))/((9/5))`
`= 1/9`
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Prove the following trigonometric identities.
`(1 + sec theta)/sec theta = (sin^2 theta)/(1 - cos theta)`
Prove the following trigonometric identities.
`cos A/(1 - tan A) + sin A/(1 - cot A) = sin A + cos A`
Prove the following trigonometric identities.
`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta)/cos theta`
Prove the following identities:
`1/(tan A + cot A) = cos A sin A`
` (sin theta - cos theta) / ( sin theta + cos theta ) + ( sin theta + cos theta ) / ( sin theta - cos theta ) = 2/ ((2 sin^2 theta -1))`
Write the value of `cosec^2 theta (1+ cos theta ) (1- cos theta).`
`If sin theta = cos( theta - 45° ),where theta " is acute, find the value of "theta` .
If cosec θ − cot θ = α, write the value of cosec θ + cot α.
If \[\sin \theta = \frac{1}{3}\] then find the value of 2cot2 θ + 2.
Prove the following identity :
tanA+cotA=secAcosecA
Prove the following identity :
`cos^4A - sin^4A = 2cos^2A - 1`
If sinA + cosA = m and secA + cosecA = n , prove that n(m2 - 1) = 2m
If secθ + tanθ = m , secθ - tanθ = n , prove that mn = 1
Prove that `sqrt((1 + sin θ)/(1 - sin θ))` = sec θ + tan θ.
Prove the following identities.
`(sin "A" - sin "B")/(cos "A" + cos "B") + (cos "A" - cos "B")/(sin "A" + sin "B")`
If sec θ = `25/7`, find the value of tan θ.
Solution:
1 + tan2 θ = sec2 θ
∴ 1 + tan2 θ = `(25/7)^square`
∴ tan2 θ = `625/49 - square`
= `(625 - 49)/49`
= `square/49`
∴ tan θ = `square/7` ........(by taking square roots)
Prove that
`(cot "A" + "cosec A" - 1)/(cot"A" - "cosec A" + 1) = (1 + cos "A")/"sin A"`
The value of 2sinθ can be `a + 1/a`, where a is a positive number, and a ≠ 1.
If tan θ + sec θ = l, then prove that sec θ = `(l^2 + 1)/(2l)`.
Complete the following activity to prove:
cotθ + tanθ = cosecθ × secθ
Activity: L.H.S. = cotθ + tanθ
= `cosθ/sinθ + square/cosθ`
= `(square + sin^2theta)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ....... тИ╡ `square`
= `1/sinθ xx 1/cosθ`
= `square xx secθ`
∴ L.H.S. = R.H.S.
