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प्रश्न
`(sin theta +cos theta )/(sin theta - cos theta)+(sin theta- cos theta)/(sin theta + cos theta) = 2/((sin^2 theta - cos ^2 theta)) = 2/((2 sin^2 theta -1))`
उत्तर
We have , `(sin theta +cos theta )/(sin theta - cos theta)+(sin theta- cos theta)/(sin theta + cos theta) `
=`((sin theta + cos theta )^2 + (sin theta - cos theta)^2) /((sin theta - cos theta )(sin theta + cos theta))`
=`(sin^2 theta + cos ^2 theta + 2 sin theta cos theta + sin^2 theta + cos^2 theta -2 sin theta cos theta)/(sin^2 theta - cos ^2 theta)`
=`(1+1)/(sin^2 theta - cos^2 theta)`
=`2/(sin^2 theta - cos^2 theta)`
Again ,` 2/(sin^2 theta - cos^2 theta)`
=`2/(sin^2 theta -(1-sin^2 theta))`
=`2/(2 sin ^2 theta -1)`
APPEARS IN
संबंधित प्रश्न
Prove the following identities:
`(i) (sinθ + cosecθ)^2 + (cosθ + secθ)^2 = 7 + tan^2 θ + cot^2 θ`
`(ii) (sinθ + secθ)^2 + (cosθ + cosecθ)^2 = (1 + secθ cosecθ)^2`
`(iii) sec^4 θ– sec^2 θ = tan^4 θ + tan^2 θ`
Prove the following trigonometric identities.
`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`
Prove the following identities:
sec2 A . cosec2 A = tan2 A + cot2 A + 2
Prove the following identities:
`1/(1 - sinA) + 1/(1 + sinA) = 2sec^2A`
Prove the following identities:
`(sinA + cosA)/(sinA - cosA) + (sinA - cosA)/(sinA + cosA) = 2/(2sin^2A - 1)`
If 2 sin A – 1 = 0, show that: sin 3A = 3 sin A – 4 sin3 A
Prove that:
`cosA/(1 + sinA) = secA - tanA`
`1/((1+ sintheta ))+1/((1- sin theta ))= 2 sec^2 theta`
If` (sec theta + tan theta)= m and ( sec theta - tan theta ) = n ,` show that mn =1
If \[\sin \theta = \frac{4}{5}\] what is the value of cotθ + cosecθ?
Prove the following identity :
`(tanθ + 1/cosθ)^2 + (tanθ - 1/cosθ)^2 = 2((1 + sin^2θ)/(1 - sin^2θ))`
If sinA + cosA = `sqrt(2)` , prove that sinAcosA = `1/2`
Prove that sin2 5° + sin2 10° .......... + sin2 85° + sin2 90° = `9 1/2`.
Prove the following identities.
`costheta/(1 + sintheta)` = sec θ – tan θ
Prove the following identities.
`(cot theta - cos theta)/(cot theta + cos theta) = ("cosec" theta - 1)/("cosec" theta + 1)`
If tan θ × A = sin θ, then A = ?
If tan θ + cot θ = 2, then tan2θ + cot2θ = ?
tan2θ – sin2θ = tan2θ × sin2θ. For proof of this complete the activity given below.
Activity:
L.H.S = `square`
= `square (1 - (sin^2theta)/(tan^2theta))`
= `tan^2theta (1 - square/((sin^2theta)/(cos^2theta)))`
= `tan^2theta (1 - (sin^2theta)/1 xx (cos^2theta)/square)`
= `tan^2theta (1 - square)`
= `tan^2theta xx square` .....[1 – cos2θ = sin2θ]
= R.H.S
If `sqrt(3) tan θ` = 1, then find the value of sin2θ – cos2θ.
Prove the following trigonometry identity:
(sinθ + cosθ)(cosecθ – secθ) = cosecθ.secθ – 2 tanθ
