Find the value of two sin 𝜃 csc 𝜃 plus 11 cos 𝜃 sec 𝜃 plus seven tan 𝜃 cot 𝜃.
To solve this problem, we need to recognize reciprocal identities. First, let’s think about ways we can rewrite cosecant, secant, and cotangent. Csc is equal to one over sin 𝜃. Sec 𝜃 is one over cos 𝜃. And cot 𝜃 is equal to one over tan 𝜃. What does this mean? Well, one over sin 𝜃 is the reciprocal of sin 𝜃. And that means cosecant is the reciprocal of sine. It means that sec 𝜃 is the reciprocal of cos 𝜃, and that cotangent is the reciprocal of tangent.
If we multiply sin 𝜃 by its reciprocal, one over sin 𝜃, we’ll get sin 𝜃 over sin 𝜃, which is one. Anything multiplied by its reciprocal is one. Two times sin 𝜃 times csc 𝜃 is equal to two times one. The same thing is true for cos 𝜃 times sec 𝜃. Cos 𝜃 times its reciprocal equals one. And so, the second term will be 11 times one. Tan 𝜃 times its reciprocal equals one. And so, our third term will be seven times one. We can simplify this expression to, say, two plus 11 plus seven, which equals 20.