If cos A-cos B-1-2 and sin A-sin B-1-4- prove that- tan A-B-2-1-2

We have, cosA+cosB=1/2 and sinA+sinB=1/4 Now, sinA+sinB/cosA+cosB =1/4/1/2 [on dividing] ⇒2sin ( A+B/2 )cos ( A B/2 )/2cos ( A+B/2 )cos ( A B/2 )=1/2 ⇒sin ( A+B ...

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Byju's Answer

Standard XII

Mathematics

Sum of Trigonometric Ratios in Terms of Their Product

If cos A+cos ...

Question

If $c o s A + c o s B = \frac{1}{2}$ and $s i n A + s i n B = \frac{1}{4}$ , prove that: $t a n (\frac{A + B}{2}) = \frac{1}{2}$

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Solution

We have,
$c o s A + c o s B = \frac{1}{2}$ and $s i n A + s i n B = \frac{1}{4}$
Now, $\frac{s i n A + s i n B}{c o s A + c o s B} = \frac{\frac{1}{4}}{\frac{1}{2}}$
[on dividing]
$\Rightarrow \frac{2 s i n (\frac{A + B}{2}) c o s (\frac{A - B}{2})}{2 c o s (\frac{A + B}{2}) c o s (\frac{A - B}{2})} = \frac{1}{2} \Rightarrow \frac{s i n (\frac{A + B}{2})}{c o s (\frac{A + B}{2})} = \frac{1}{2} \Rightarrow t a n (\frac{A + B}{2}) = \frac{1}{2}$

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If cosA+cosB=12 and sinA+sinB=14, prove that: tan(A+B2)=12

We have, cosA+cosB=12 and sinA+sinB=14 Now, sinA+sinBcosA+cosB=1412 [on dividing] ⇒2sin(A+B2)cos(A−B2)2cos(A+B2)cos(A−B2)=12⇒sin(A+B2)cos(A+B2)=12⇒tan(A+B2)=12

If $c o s A + c o s B = \frac{1}{2}$ and $s i n A + s i n B = \frac{1}{4}$ , prove that: $t a n (\frac{A + B}{2}) = \frac{1}{2}$