If A- B- C are in A-P- then sin A-sin C-cos C-cosA-

The correct option is D cot B If A, B, C are in A.P. B A=C B Or. 2B=A+C=sinA sinC/cosC cosA =2sin ( A C/2 )cos ( A+C/2 )/ 2sin ( C+A/2 )sin ( C A/2 ) [ ∵ sin ...

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Byju's Answer

Standard XII

Mathematics

Sum of Trigonometric Ratios in Terms of Their Product

If A, B, C ar...

Question

If A, B, C are in A.P., then $\frac{s i n A - s i n C}{c o s C - c o s A} =$

tanB

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cot B

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tan 2B

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None of these

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Solution

The correct option is B
cot B

If A, B, C are in A.P.
B-A=C-B
Or. $2 B = A + C = \frac{s i n A - s i n C}{c o s C - c o s A} = \frac{2 s i n (\frac{A - C}{2}) c o s (\frac{A + C}{2})}{- 2 s i n (\frac{C + A}{2}) s i n (\frac{C - A}{2})} [∵ s i n A - s i n B = 2 s i n (\frac{A - B}{2}) c o s (\frac{A + B}{2})] a n d c o s A - c o s B = - 2 s i n (\frac{A + B}{2}) c o s (\frac{A - B}{2}) = \frac{s i n (\frac{A - C}{2}) c o s (\frac{A + C}{2})}{s i n (\frac{A + C}{2}) s i n (\frac{A - C}{2})} = \frac{c o s (\frac{A + C}{2})}{s i n (\frac{A + C}{2})} = \frac{c o s B}{s i n B} = c o t B$

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If A, B, C are in A.P., then sinA−sinCcosC−cosA=

The correct option is B cot B If A, B, C are in A.P. B-A=C-B Or. 2B=A+C=sinA−sinCcosC−cosA=2sin(A−C2)cos(A+C2)−2sin(C+A2)sin(C−A2)[∵sinA−sinB=2sin(A−B2)cos(A+B2)]and cosA−cosB=−2sin(A+B2)cos(A−B2)=sin(A−C2)cos(A+C2)sin(A+C2)sin(A−C2)=cos(A+C2)sin(A+C2)=cosBsinB=cotB

If A, B, C are in A.P., then $\frac{s i n A - s i n C}{c o s C - c o s A} =$