Answer:
Refer below.
Step-by-step explanation:
To prove the given trigonometric identity, we need to start by expressing each trigonometric function in terms of sine and cosine. This will allow us to manipulate the equation and show that both sides are equivalent.
We are given:
\(\csc(A)\sec(A)\tan(A)=\sec^2(A)\)
We will manipulate the more complex side. First, express each function in terms of sine and cosine:
- csc(A) = 1/sin(A)
- sec(A) = 1/cos(A)
- tan(A) = sin(A)/cos(A)
Now substitute these expressions into the left side of the given identity:
\(\Longrightarrow \left(\dfrac{1}{\sin(A)}\right)\left(\dfrac{1}{\cos(A)}\right)\left(\dfrac{\sin(A)}{\cos(A)}\right)=\sec^2(A)\)
Combine the fractions:
\(\Longrightarrow \dfrac{\sin(A)}{\sin(A)\cos^2(A)}=\sec^2(A)\)
Cancel the sin(A) terms:
\(\Longrightarrow \dfrac{1}{\cos^2(A)}=\sec^2(A)\)
We know that 1/cos(A) = sec(A). Thus,
\(\therefore \sec^2(A)=\sec^2(A)\)
