In a right triangle, if angles $b$ and $c$ are the two non-right angles, what is the value of the product $\sin b\times \tan c$ and the product $\sin c\times \tan b$?

• A. Both products equal $1$
• B. The first product is $0$ and the second is $1$
• C. Both products equal $0$
• D. The first product is $1$ and the second is $0$

Answer: (A)

Verified step by step guidance

1. Recall that in a right triangle, the two non-right angles \( b \) and \( c \) satisfy the relationship \( b + c = 90^\circ \) or \( b + c = \frac{\pi}{2} \) radians.
2. Use the complementary angle identity for sine and cosine: since \( c = 90^\circ - b \), we have \( \sin(c) = \sin(90^\circ - b) = \cos(b) \) and similarly \( \sin(b) = \cos(c) \).
3. Express the tangent functions in terms of sine and cosine: \( \tan(c) = \frac{\sin(c)}{\cos(c)} \) and \( \tan(b) = \frac{\sin(b)}{\cos(b)} \).
4. Substitute these into the products: First product: \( \sin(b) \times \tan(c) = \sin(b) \times \frac{\sin(c)}{\cos(c)} \). Second product: \( \sin(c) \times \tan(b) = \sin(c) \times \frac{\sin(b)}{\cos(b)} \).
5. Replace \( \sin(b) \) and \( \sin(c) \) using the complementary angle identities and simplify the expressions to show that both products equal 1.