-If a-sin-bcos-c-then prove that a-cos-bsin-a-2-b-2-c-2-

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Implicit Differentiation

If a sinθ+bco...

Question

If a \operatorname{sinθ+b\operatorname{cosθ=c, then prove that a \operatorname{cosθ-b\operatorname{sinθ=\sqrt{a^2+b^2-c^2

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a, b, c

a^{2} = b^{2} + c^{2} - 2 b c \sqrt{1 - a^{2}}, b^{2} = c^{2} + a^{2} - 2 c a \sqrt{1 - b^{2}}, c^{2} = a^{2} + b^{2} - 2 a b \sqrt{1 - c^{2}}

a = c \sqrt{1 - b^{2}} + b \sqrt{1 - c^{2}}

\frac{a sin (B - C)}{b^{2} - c^{2}} = \frac{b sin (C - A)}{c^{2} - a^{2}} = \frac{c sin (A - B)}{a^{2} - b^{2}}

a^{2} + b^{2} + c^{2} - a b - b c - c a = \frac{1}{2} [{(b - c)}^{2} + {(c - a)}^{2} + {(a - b)}^{2}]

\frac{b^{2} + c^{2}}{b + c} + \frac{c^{2} + a^{2}}{c + a} + \frac{a^{2} + b^{2}}{a + b} \geq a + b + c

\frac{(b^{2} + c^{2})}{3 b c} + \frac{(c + a)^{2}}{3 a c} + \frac{(a + b)^{2}}{3 a b} = \frac{1}{3}

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