sin theta by cot theta + cosec theta -sin theta by cot theta – cosectheta=2​

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sin theta by cot theta + cosec theta -sin theta by cot theta – cosectheta=2​

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Mia 3 months 2021-10-15T16:34:27+00:00 2 Answers 0 views 0

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    0
    2021-10-15T16:35:27+00:00

    \underline\mathfrak\pink{Questions:-}

    \: \: \: \: \: \therefore \: \: \orange{\frac{Sin \theta}{Cot \theta \: + \: Cosec \theta} \: - \: \frac{Sin \theta}{Cot \theta \: - \: Cosec \theta} \: \: = \: \: {2}}

    \large\underline\mathfrak\pink{Prove:-}

    \: \: \: \: \: \underline\red{LHS}

    \: \: \: \: \: \leadsto \: \: \green{\frac{Sin \theta}{Cot \theta \: + \: Cosec \theta} \: - \: \frac{Sin \theta}{Cot \theta \: - \: Cosec \theta}}

    \: \: \: \: \: \leadsto \: \: \green{\frac{Sin \theta {(Cot \theta \: - \: Cosec \theta)} \: - \: Sin \theta \: {(Cot \theta \: + \: Cosec \theta)}}{{(Cot \theta \: + \: Cosec \theta)} \: {(Cot \theta \: - \: Cosec \theta)}}}

    \: \: \: \: \: \leadsto \: \: \green{\frac{Sin \theta {(Cot \theta \: - \: Cosec \theta \: - \: Cot \theta \: - \: Cosec \theta)}}{({Cot}^{2} \theta \: - \: {Cosec}^{2} \theta)}}

    \: \: \: \: \: \leadsto \: \: \green{\frac{Sin \theta {({-2} \: Cosec \theta)}}{ - \: ({Cot}^{2} \theta \: - \: {Cosec}^{2} \theta)}}

    \: \: \: \: \: \leadsto \: \: \green{\frac{ {2} \: Sin \theta {(\frac{1}{Sin \theta})}}{1}}

    \: \: \: \: \: \leadsto \: \: \green{{2} \: Sin \theta \: \times \: \frac{1}{Sin \theta}}

    \: \: \: \: \: \leadsto \: \: \green{2}

    \: \: \: \: \: \: \: \: \underline\red{Proved}

    \large\underline\pink{More \: Information:-}

    • \: \: \: \: \: {Cos}^{2} \theta \: + \: {Sin}^{2} \theta \: \: = \: \: {1}
    • \: \: \: \: \: {1} \: + \: {tan}^{2} \theta \: \: = \: \: {Sec}^{2} \theta
    • \: \: \: \: \: {1} \: + \: {Cot}^{2} \theta \: \: = \: \: {Cosec}^{2} \theta
    • \: \: \: \: \: tan \: {(x \: + \:  y)} \: \: = \: \: \frac{tan \: x \: + \: tan \: y}{{1} \: - \: tan \: x \: tan \: y}
    • \: \: \: \: \: tan \: {(x \: - \:  y)} \: \: = \: \: \frac{tan \: x \: - \: tan \: y}{{1} \: + \: tan \: x \: tan \: y}
    • \: \: \: \: \: Sin \: {2x} \: \: = \: \: {2} \: Sin \: x \: Cos \: x \: \: = \: \: \frac{{2} \: tan \: x}{{1} \: + \: {tan}^{2} \: x}
    • \: \: \: \: \: tan \: {2x} \: \: = \: \: \frac{{2} \: tan \: x}{{1} \: - \: {tan}^{2} \: x}
    • \: \: \: \: \: Cos \: x \: + \: Cos \: y \: \: = \: \: {2} \: Cos \: \frac{x \: + \: y}{2} \: Cos \: \frac{x \: - \: y}{2}
    • \: \: \: \: \: Sin \: x \: + \: Sin \: y \: \: = \: \: {2} \: Sin \: \frac{x \: + \: y}{2} \: Sin \: \frac{x \: - \: y}{2}
    • \: \: \: \: \: Sin \: x \: - \: Sin \: y \: \: = \: \: {2} \: Cos \: \frac{x \: + \: y}{2} \: Sin \: \frac{x \: - \: y}{2}

    __________________________

    0
    2021-10-15T16:36:03+00:00

    (sin theta/cot theta+cosec theta) – (sin theta/cot theta-cosec theta =2 [we have to prove this]

    LHS= Sin theta[1/cot theta+cosec theta – 1/cot theta-cosec theta]

    sin theta[cot theta -cosec theta – cot theta -cosec theta/ (cot theta+cosec theta) (cot theta- cosec theta)]

    cutting cot theta

    sin theta[-2 cosec theta /cot^2 theta – cosec^2 theta]

    sin theta(-2 cosec theta/-1)

    (using indentity= 1+cot^2 theta= cosec^2 theta)

    sin theta × 2 × 1/ sin theta =2

    R.H.S =2

    hence proved

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