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Dmitriy789 [7]
3 years ago
6

Tan^2theta - Sin^2theta = tan^2theta. sin^2theta explain this step​

Mathematics
1 answer:
denis23 [38]3 years ago
8 0

tan²(<em>θ</em>) - sin²(<em>θ</em>) = sin²(<em>θ</em>)/cos²(<em>θ</em>) - sin²(<em>θ</em>)

-- because tan(<em>θ</em>) = sin(<em>θ</em>)/cos(<em>θ</em>) by definition of tangent --

… = sin²(<em>θ</em>) (1/cos²(<em>θ</em>) - 1)

-- we pull out the common factor of sin²(<em>θ</em>) from both terms --

… = sin²(<em>θ</em>) (1/cos²(<em>θ</em>) - cos²(<em>θ</em>)/cos²(<em>θ</em>))

-- because <em>x</em>/<em>x</em> = 1 (so long as <em>x</em> ≠ 0) --

… = sin²(<em>θ</em>) ((1 - cos²(<em>θ</em>))/cos²(<em>θ</em>))

-- we simply combine the fractions, which we can do because of the common denominator of cos²(<em>θ</em>) --

… = sin²(<em>θ</em>) (sin²(<em>θ</em>)/cos²(<em>θ</em>))

-- due to the Pythagorean identity, sin²(<em>θ</em>) + cos²(<em>θ</em>) = 1 --

… = sin²(<em>θ</em>) tan²(<em>θ</em>)

-- again, by definition of tan(<em>θ</em>) --

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