Question

sin(a-b)/cosacosb=tana-tanb

Trigonometric Functions
ANSWERED
asked 2021-06-10
\(\displaystyle\frac{{\sin{{\left({a}-{b}\right)}}}}{{\cos{{a}}}}{\cos{{b}}}={\tan{{a}}}-{\tan{{b}}}\)

Expert Answers (1)

2021-06-11

Work using the left side. Use the sine of a difference: \(\displaystyle{\sin{{\left({x}-{y}\right)}}}={\sin{{x}}}{\cos{{y}}}-{\cos{{x}}}{\sin{{y}}}\)
\(\frac{\sin(a-b)}{\cos a\cos b}=\frac{\sin a\cos b-\cos a\sin b}{\cos a\cos b}\)
Separate as:
\(\frac{\sin(a-b)}{\cos a\cos b}=\left(\frac{\sin a\cos b}{\cos a\cos b}\right)-\left(\frac{\cos a\sin b}{\cos a\cos b}\right)\)
Cancel common factors: \(\frac{\sin(a-b)}{\cos a\cos b}=\left(\frac{\sin a}{cos a}\right)-\left(\frac{\sin b}{\cos b}\right)\)
Use the quotient identity: \(\displaystyle{\tan{{x}}}=\frac{{\sin{{x}}}}{{\cos{{x}}}}\)
\(\frac{\sin(a-b)}{\cos a\cos b}=\tan a-\tan b\)

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