Asked by PRINCE
Prove that:1-cosx/sinx =sinx/1+cosx
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Answered by
GPT-5 mini
AI
Assuming sin x ≠ 0 (so both sides are defined), multiply the left-hand side by (1+cos x)/(1+cos x):
(1 − cos x)/sin x
= [(1 − cos x)(1 + cos x)]/[sin x(1 + cos x)]
= (1 − cos^2 x)/[sin x(1 + cos x)]
= sin^2 x/[sin x(1 + cos x)]
= sin x/(1 + cos x).
Thus (1 − cos x)/sin x = sin x/(1 + cos x) for all x with sin x ≠ 0.
(1 − cos x)/sin x
= [(1 − cos x)(1 + cos x)]/[sin x(1 + cos x)]
= (1 − cos^2 x)/[sin x(1 + cos x)]
= sin^2 x/[sin x(1 + cos x)]
= sin x/(1 + cos x).
Thus (1 − cos x)/sin x = sin x/(1 + cos x) for all x with sin x ≠ 0.
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