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рдкреНрд░рд╢реНрди
`tan theta/(1+ tan^2 theta)^2 + cottheta/(1+ cot^2 theta)^2 = sin theta cos theta`
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рдЙрддреНрддрд░
ЁЭР┐ЁЭР╗ЁЭСЖ = `(tan theta)/(1+tan^2 theta )^2 +( cot theta )/(1+cot^2 theta)^2`
=`tan theta/ ((sec^2 theta)^2) + cot theta/((cosec^2 theta) ^2)`
=`tan theta / sec^4 theta + cottheta/(cosec^4 theta)`
=`sin theta/cos theta xx cos^4 theta + cos theta/sin theta xx sin ^4 theta`
=` sin theta cos ^3 theta + cos theta sin ^3 theta`
=`sin theta cos theta ( cos^2 theta + sin ^2 theta)`
=`sin theta cos theta`
= RHS
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рд╕рдВрдмрдВрдзрд┐рдд рдкреНрд░рд╢реНрди
Prove that: `(1 – sinθ + cosθ)^2 = 2(1 + cosθ)(1 – sinθ)`
Prove the following trigonometric identities.
`tan theta + 1/tan theta = sec theta cosec theta`
Prove the following trigonometric identities.
`sin A/(sec A + tan A - 1) + cos A/(cosec A + cot A + 1) = 1`
Prove the following identities:
`(1+ sin A)/(cosec A - cot A) - (1 - sin A)/(cosec A + cot A) = 2(1 + cot A)`
If x = r sin A cos B, y = r sin A sin B and z = r cos A, then prove that : x2 + y2 + z2 = r2
Prove the following identities:
`1 - sin^2A/(1 + cosA) = cosA`
Prove that
`cot^2A-cot^2B=(cos^2A-cos^2B)/(sin^2Asin^2B)=cosec^2A-cosec^2B`
`cosec theta (1+costheta)(cosectheta - cot theta )=1`
Write the value of `(1 - cos^2 theta ) cosec^2 theta`.
Write the value of `(1 + cot^2 theta ) sin^2 theta`.
Define an identity.
Prove the following identity :
secA(1 - sinA)(secA + tanA) = 1
Given `cos38^circ sec(90^circ - 2A) = 1` , Find the value of <A
Prove that:
`sqrt(( secθ - 1)/(secθ + 1)) + sqrt((secθ + 1)/(secθ - 1)) = 2cosecθ`
Prove that `sqrt(2 + tan^2 θ + cot^2 θ) = tan θ + cot θ`.
If A = 60°, B = 30° verify that tan( A - B) = `(tan A - tan B)/(1 + tan A. tan B)`.
If tan A + sin A = m and tan A − sin A = n, then show that `m^2 - n^2 = 4 sqrt (mn)`.
Prove that `((1 + sin θ - cos θ)/( 1 + sin θ + cos θ))^2 = (1 - cos θ)/(1 + cos θ)`.
Prove that sec2θ + cosec2θ = sec2θ × cosec2θ
Show that, cotθ + tanθ = cosecθ × secθ
Solution :
L.H.S. = cotθ + tanθ
= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
L.H.S. = R.H.S
∴ cotθ + tanθ = cosecθ × secθ
