Question

√1+Sin A÷√1-Sin A=(tanΠ\4+A\2)

Answer 1

To prove :-

$\tt \:  \sqrt{\dfrac{1 + sinA}{1 - sinA} } = tan \bigg( \dfrac{\pi}{4} + \dfrac{A}{2} \bigg)$

Identities Used :

$\red{\tt \: (1). \: cos( \dfrac{\pi}{2} - x) = sinx}$

$\red{\tt \:  (2). \: \: 1 - cos2x = 2 {sin}^{2} x}$

$\blue{\tt \:  (3). \: \: 1 + cos2x = {2cos}^{2} x}$

$\blue{\tt \:  (4). \: \: tan(\dfrac{\pi}{2} - x) = \: cotx}$

Solution:-

☆ Consider LHS

$\tt \:  \longrightarrow \sqrt{\dfrac{1 + sinA}{1 - sinA} }$

$\tt \:  \longrightarrow \: = \sqrt{\dfrac{1 + cos(\dfrac{\pi}{2} - A)}{1 - cos(\dfrac{\pi}{2} - A)} }$

$\tt \:  \longrightarrow \: Let \: \dfrac{\pi}{2} - A = y$

$\tt\implies \:\sqrt{\dfrac{1 + cos(\dfrac{\pi}{2} - A)}{1 - cos(\dfrac{\pi}{2} - A)} } = \sqrt{\dfrac{1 + cosy}{1 - cosy} }$

$\tt \:  \longrightarrow \: = \: \sqrt{\dfrac{2 {cos}^{2}\dfrac{y}{2} }{ {2sin}^{2} \dfrac{y}{2} } }$

$\tt \:  \longrightarrow \: = \: \dfrac{cos\dfrac{y}{2} }{sin\dfrac{y}{2} }$

$\tt \:  \longrightarrow \: = cot\dfrac{y}{2}$

$\tt \:  \longrightarrow \: = cot\dfrac{1}{2} (\dfrac{\pi}{2} - A)$

$\tt \:  \longrightarrow \: = \: cot \bigg(\dfrac{\pi}{4} - \dfrac{A}{2} \bigg)$

$\tt \:  \longrightarrow \: = \: tan \bigg(\dfrac{\pi}{2} - \dfrac{\pi}{4} + \dfrac{A}{2} \bigg)$

$\tt \:  \longrightarrow \: = \: tan \bigg(\dfrac{\pi}{4} + \dfrac{A}{2} \bigg)$

$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

$\large \red{\bf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: Explore \: \: \: more } ✍$

Trigonometry Formulas

sin(−θ) = −sin θ

cos(−θ) = cos θ

tan(−θ) = −tan θ

cosec(−θ) = −cosecθ

sec(−θ) = sec θ

cot(−θ) = −cot θ

Product to Sum Formulas

sin x sin y = 1/2 [cos(x–y) − cos(x+y)]

cos x cos y = 1/2[cos(x–y) + cos(x+y)]

sin x cos y = 1/2[sin(x+y) + sin(x−y)]

cos x sin y = 1/2[sin(x+y) – sin(x−y)]

Sum to Product Formulas

sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2]

sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2]

cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2]

cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]

Sum or Difference of angles

cos (A + B) = cos A cos B – sin A sin B

cos (A – B) = cos A cos B + sin A sin B

sin (A+B) = sin A cos B + cos A sin B

sin (A -B) = sin A cos B – cos A sin B

tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)]

tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]

cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)]

cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)]

cos(A+B) cos(A–B)=cos^2A–sin^2B=cos^2B–sin^2A

sin(A+B) sin(A–B) = sin^2A–sin^2B=cos^2B–cos^2A

Multiple and Submultiple angles

sin2A = 2sinA cosA = [2tan A /(1+tan²A)]

cos2A = cos²A–sin²A = 1–2sin²A = 2cos²A–1= [(1-tan²A)/(1+tan²A)]

tan 2A = (2 tan A)/(1-tan²A)