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प्रश्न
Prove the following identities.
`(cot theta - cos theta)/(cot theta + cos theta) = ("cosec" theta - 1)/("cosec" theta + 1)`
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उत्तर
`(cot theta - cos theta)/(cot theta + cos theta) = ("cosec" theta - 1)/("cosec" theta + 1)`
L.H.S = `(cot theta - cos theta)/(cot theta + cos theta)`
= `cos theta/sin theta - cos theta ÷ cos theta/sin theta + cos theta`
= `(cos theta - sin theta cos theta)/sin theta ÷ (cos theta + sin theta cos theta)/sin theta`
= `(cos theta(1 - sin theta))/sin theta ÷ (cos theta(1 + sin theta))/sin theta`
= `(cos theta(1 - sin theta))/sin theta xx sin theta/(cos theta(1 + sin theta))`
= `(1 - sin theta)/(1 + sin theta)`
R.H.S = `("cosec" - 1)/("cosec"+1)`
= `1/sin theta - 1 ÷ 1/sin theta+ 1`
= `(1 - sin theta)/sin theta ÷ (1 + sin theta)/sin theta`
= `(1 - sin theta)/sin theta xx sin theta/(1 + sin theta)`
= `(1 - sin theta)/(1 + sin theta)`
R.H.S = L.H.S ⇒ `(cot theta - cos theta)/(cot theta + cos theta) = ("cosec" theta - 1)/("cosec" theta + 1)`
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संबंधित प्रश्न
if `x/a cos theta + y/b sin theta = 1` and `x/a sin theta - y/b cos theta = 1` prove that `x^2/a^2 + y^2/b^2 = 2`
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`cot theta/((cosec theta + 1) )+ ((cosec theta +1 ))/ cot theta = 2 sec theta `
Show that none of the following is an identity:
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Write True' or False' and justify your answer the following :
The value of \[\cos^2 23 - \sin^2 67\] is positive .
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Prove the following identity :
`(cotA + tanB)/(cotB + tanA) = cotAtanB`
Prove the following identity :
`cosA/(1 - tanA) + sin^2A/(sinA - cosA) = cosA + sinA`
Prove that sec2θ + cosec2θ = sec2θ × cosec2θ.
tan2θ – sin2θ = tan2θ × sin2θ. For proof of this complete the activity given below.
Activity:
L.H.S. = `square`
= `square (1 - (sin^2θ)/(tan^2θ))`
= `tan^2θ (1 - square/((sin^2θ)/(cos^2θ)))`
= `tan^2θ (1 - (sin^2θ)/1 xx (cos^2θ)/square)`
= `tan^2θ (1 - square)`
= `tan^2θ xx square` ...[1 – cos2θ = sin2θ]
= R.H.S.
