# Chapter 5 - Trigonometric Identities - Section 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 228: 66

$$\frac{\sin(s+t)}{\cos s\cos t}=\tan s+\tan t$$ We tackle the left side first to find out that this is an identity.

#### Work Step by Step

$$\frac{\sin(s+t)}{\cos s\cos t}=\tan s+\tan t$$ We start from the left side. $$X=\frac{\sin(s+t)}{\cos s\cos t}$$ $\sin(s+t)$ would be expanded using the sine sum identity: $$\sin(s+t)=\sin s\cos t+\cos s\sin t$$ So, $$X=\frac{\sin s\cos t+\cos s\sin t}{\cos s\cos t}$$ Now we separate $X$ into 2 fractions: $$X=\frac{\sin s\cos t}{\cos s \cos t}+\frac{\cos s\sin t}{\cos s\cos t}$$ $$X=\frac{\sin s}{\cos s}+\frac{\sin t}{\cos t}$$ Here recall that $\frac{\sin\theta}{\cos\theta}=\tan\theta$. $$X=\tan s+\tan t$$ Therefore the equation has been proved. It is an identity.

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