How do I prove this equation? 1/sec theta+tan theta = (1-sin theta)/cos theta?

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  • 4 weeks ago

    Ambiguity alert!

    Did you mean (A) (1/sec theta) + (tan theta) = (1-sin theta)/(cos theta)

    or (B) 1/[(sec theta)+(tan theta)] = (1-sin theta)/(cos theta)

    I will assume you meant (B)

    (sec theta)+(tan theta) = 1/(cos theta) + (sin theta)/(cos (theta)) = (1 + sin theta)/(cos theta)

    Therefore, 1/[(sec theta)+(tan theta)] = (cos theta)/(1 + sin theta)

    This makes the equation (cos theta)/(1 + sin theta) = (1 - sin theta)/(cos theta)

    Multiply both sides by (1 + sin theta)(cos theta) to get

    (cos theta)² = (1 - sin theta)(1 + sin theta) = 1 - (sin theta)², which becomes

    (sin theta)² + (cos theta)² = 1 which is true.

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