Question

cos7A = cosA - sin4A (Find the value of A if 0<A<180)​

Answer 1

Answer:

The value of A is 10 degrees

step-by-step explanation:

$\text{Given:}$

$cos\,7A=cos\,A-sin\,4A$

$\implies\,sin\,4A=cos\,A-cos\,7A$

Using

$\boxed{\bf\;cos\,C-cos\,D=-2\;sin(\frac{C+D}{2})\;sin(\frac{C-D}{2})}$

we get

$sin\,4A=-2\;sin(\frac{A+7A}{2})\;sin(\frac{A-7A}{2})$

$sin\,4A=-2\;sin\,4A\;sin(-3A)$

Using

$\boxed{\;sin(-\theta)=-\;sin\theta}$

we get

$1=2\;sin\,3A$

$sin\,3A=\frac{1}{2}$

$sin\,3A=sin\,30^{\circ}$

$\implies\;3A=30^{\circ}$

$\implies\boxed{\bf\,A=10^{\circ}}$