Solve 1-tan-1-sin 2-1-tan-

Replace sin 2θ by 2tanθ /(1+tan^2θ) 1+tanθ1 tanθ=1+2tanθ1+tan^2θ=(1+tanθ)^21+tan^2θ or (1+tanθ)(2tan^2θ)=0 ∴tanθ =0, tanθ = 1=tan( π/4 ...

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Trigonometric Ratios of Allied Angles

Solve 1-tan...

Question

Solve $(1 - tan θ) (1 + sin 2 θ) = 1 + tan θ$ .

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Solution

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(1 - tan θ) (1 + sin 2 θ) = 1 + tan θ

θ \in [0, 2 π)]

{sin}^{2} θ tan θ + {cos}^{2} θ cot θ - sin 2 θ = 1 + tan θ + cot θ

[\begin{matrix} 1 & - t a n θ t a n θ & 1 \end{matrix}]

{[\begin{matrix} 1 & - t a n θ t a n θ & 1 \end{matrix}]}^{- 1} = {[\begin{matrix} c o s a & - s i n a s i n a & c o s a \end{matrix}]}^{- 1}

a =

[\begin{matrix} 1 & - tan θ tan θ & 1 \end{matrix}] {[\begin{matrix} 1 & tan θ - tan θ & 1 \end{matrix}]}^{- 1} = [\begin{matrix} a & - b b & a \end{matrix}]

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

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