Let z=tan theta and theta is acute. Show that cos(2t)=(1-z^2/1+z^2) and sin(2t)=2z/1+z^2

Answer 1

(1 - z²) / (1 + z²) = (1 - tan²θ) / (1 + tan²θ) = [1 - sin²θ/cos²θ] / sec²θ

= [(cos²θ - sin²θ) / cos²θ] ÷ [1 / cos²θ]

= [cos(2θ) / cos²θ] [ cos²θ / 1] = cos(2θ)

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2z / (1 + z²) = 2tanθ / (1 + tan²θ) = 2(sinθ/cosθ) / sec²θ)

= 2(sinθ/cosθ) / [1 / cos²θ] = 2(sinθ/cosθ)(cos²θ) = 2sinθcosθ = sin(2θ)

Answer 2

Let z=tan theta and theta is acute. Show that cos(2t)=(1-z^2/1+z^2) and sin(2t)=2z/1+z^2

Is a set of ( ) to be within (1-z^2...........)?