Show that cos Aa-cos Bb-cos Cc-a2-b2-c22abc-

Now, cos Aa+cos Bb+cos Cc =b^2+c^2 a^22bca+a^2+c^2 b^22acb+a^2+b^2 c^22abc #160; #160; [ Using the values of cos A,cos B,cos C ] =b^2+c^ ...

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Standard XII

Mathematics

Higher Order Derivatives

Show that c...

Question

Show that $\frac{cos A}{a} + \frac{cos B}{b} + \frac{cos C}{c} = \frac{a^{2} + b^{2} + c^{2}}{2 a b c}$ .

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Solution

Now,

$\frac{cos A}{a} + \frac{cos B}{b} + \frac{cos C}{c}$

$= \frac{\frac{b^{2} + c^{2} - a^{2}}{2 b c}}{a} + \frac{\frac{a^{2} + c^{2} - b^{2}}{2 a c}}{b} + \frac{\frac{a^{2} + b^{2} - c^{2}}{2 a b}}{c}$ [ Using the values of $cos A, cos B, cos C$ ]

$= \frac{b^{2} + c^{2} - a^{2}}{2 a b c} + \frac{a^{2} + c^{2} - b^{2}}{2 a b c} + \frac{a^{2} + b^{2} - c^{2}}{2 a b c}$

$= \frac{a^{2} + b^{2} + c^{2}}{2 a b c}$ .

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\frac{c o s A}{a} + \frac{c o s B}{b} + \frac{c o s C}{c} = \frac{a^{2} + b^{2} + c^{2}}{2 a b c}

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\frac{cos A}{a} + \frac{cos B}{b} + \frac{cos C}{c} = \frac{a^{2} + b^{2} + c^{2}}{2 a b c}

Q. Prove that

$\frac{c o s A}{c c o s B + b c o s C} + \frac{c o s B}{a c o s C + c c o s A} + \frac{c o s C}{a c o s B + b c o s A} = \frac{a^{2} + b^{2} + c^{2}}{2 a b c}$

Q. Prove that:

\frac{a^{2} - b^{2}}{cos A + cos B} + \frac{b^{2} - c^{2}}{cos B + cos C} + \frac{c^{2} - a^{2}}{cos C + cos A} = 0

Q. Prove that:

\frac{b^{2} - c^{2}}{cos B + cos C} + \frac{c^{2} - a^{2}}{cos C + cos A} + \frac{a^{2} - b^{2}}{cos A + cos B} = 0.

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Show that cosAa+cosBb+cosCc=a2+b2+c22abc.

Now,cosAa+cosBb+cosCc=b2+c2−a22bca+a2+c2−b22acb+a2+b2−c22abc [ Using the values of cosA,cosB,cosC]=b2+c2−a22abc+a2+c2−b22abc+a2+b2−c22abc=a2+b2+c22abc.

Show that $\frac{cos A}{a} + \frac{cos B}{b} + \frac{cos C}{c} = \frac{a^{2} + b^{2} + c^{2}}{2 a b c}$ .