Tan 2- - 1 cos2 - - 1 - 0

L.H.S = tan2 #160; #952; #8211; (1/cos2 #160; #952;) + 1= tan2 #160; #952; sec2 #160; #952; + 1 [since, 1/cos #952; = sec #952;] #1 ...

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General Solution of Trigonometric Equation

tan 2θ – 1 co...

Question

$t a n 2 θ - (\frac{1}{c o s 2 θ}) + 1 = 0$

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Solution

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(i) \cot^{2} θ - \frac{1}{\sin^{2} θ} = - 1 (ii) \tan^{2} θ - \frac{1}{\cos^{2} θ} = - 1 (iii) \cos^{2} θ + \frac{1}{(1 + \cot^{2} θ)} = 1

({cos}^{2} θ - 1) ({cos}^{2} θ + 1) + 1 =

[\frac{1 + {tan}^{2} θ}{1 + {cot}^{2} θ}] = {[\frac{1 - tan θ}{1 - cot θ}]}^{2} = {tan}^{2} θ

θ = 30^{\circ}

tan 2 θ = \frac{2 tan θ}{1 - {tan}^{2} θ}

tan 2 θ = \frac{4 tan θ}{1 + {tan}^{2} θ}

cos 2 θ = \frac{1 - {tan}^{2} θ}{1 + {tan}^{2} θ}

cos 3 θ = 4 {cos}^{3} θ - 3 cos θ

y = c o s^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}); 0 < x < 1

x = t a n θ

y = c o s^{- 1} (c o s 2 θ) = 2 θ

y = 2 t a n^{- 1} x

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