Question

Tan theta +secthetab-1/tanTheta-sectheta+1=1+sintheta/costheta

Answer 1

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"Good question".

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Answer 2

Answer:

$\frac{Tan\theta + Sec\theta - 1}{Tan\theta - Sec\theta + 1} = \frac{1 +Sin\theta}{Cos\theta}$

Step-by-step explanation:

Tan theta +secthetab-1/tanTheta-sectheta+1=1+sintheta/costheta

$\frac{Tan\theta + Sec\theta - 1}{Tan\theta - Sec\theta + 1} \\\\= \frac{\frac{Sinn\theta}{Cos\theta} + \frac{1}{Cos\theta}- 1}{\frac{Sinn\theta}{Cos\theta} - \frac{1}{Cos\theta} + 1} \\\\= \frac{Sin\theta + 1 - Cos\theta}{Sin\theta - 1 + Cos\theta} \\\\= \frac{Sin\theta + (1 - Cos\theta)}{Sin\theta - (1 - Cos\theta)}\\\\multiplying\: \&\: dividing\: by\: Sin\theta + (1 - Cos\theta)$

$Numerator = Sin^2\theta + 1 + Cos^2\theta - 2cos\theta + 2Sin\theta - 2Sin\theta Cos\theta\\\\Using \:Sin^2\theta + Cos^2\theta = 1\\\\\implies Numerator = 2 - 2cos\theta + 2Sin\theta - 2Sin\theta Cos\theta\\\\\implies Numerator = 2(1 - Cos\theta) + 2Sin\theta (1-Cos\theta)\\\\\implies Numerator = (2 + 2Sin\theta)(1-Cos\theta)\\\\\implies Numerator = 2(1+Sin\theta)(1-Cos\theta)$

$Denominator = Sin^2\theta - (1 + Cos^2\theta - 2Cos\theta) \\\\\implies Denominator = Sin^2\theta - 1 - Cos^2\theta + 2Cos\theta)\\\\Using\: Sin^2\theta - 1 = -Cos^2\theta\\\\\implies Denominator = -Cos^2\theta - Cos^2\theta + 2Cos\theta\\\\\implies Denominator = - 2Cos^2\theta + 2Cos\theta\\\\\implies Denominator = 2Cos\theta(1 - Cos\theta)$

LHS = Numerator / Denominator

$=\frac{2(1+Sin\theta)(1-Cos\theta)}{2Cos\theta)(1-Cos\theta)} \\\\Cancelling (1 - Cos\theta) \: \& \: 2\: from\: Numerator\: \& \: Denominator\\\\= \frac{1+Sin\theta}{Cos\theta}$

= RHS

$\frac{Tan\theta + Sec\theta - 1}{Tan\theta - Sec\theta + 1} = \frac{1 +Sin\theta}{Cos\theta}$

QED