Verify the identity. Show your work.
cot θ ∙ sec θ = csc θ
cot θ ∙ sec θ = csc θ
Note that cot θ = cos θ / sin θ, and sec θ = 1 / cos θ:
cos θ / sin θ ∙ 1 / cos θ = csc θ
1 / sin θ = csc θ
Note that 1 / sin θ = csc θ. Thus:
csc θ = csc θ
Thank you !
To verify the given identity cot θ ∙ sec θ = csc θ, we will start with the left side of the equation and simplify it using trigonometric identities.
Recall the definitions of the trigonometric functions:
cot θ = cos θ / sin θ ---- (1)
sec θ = 1 / cos θ ------ (2)
csc θ = 1 / sin θ ------ (3)
Now, substitute equations (1) and (2) into the left side of the original equation:
cot θ ∙ sec θ = (cos θ / sin θ) ∙ (1 / cos θ)
Simplifying further:
cot θ ∙ sec θ = (cos θ / sin θ) ∙ (1 / cos θ)
= cos θ / (sin θ ∙ cos θ)
Now, we know the identity sin θ ∙ cos θ = 1/2 sin 2θ. Substitute this into the equation:
cos θ / (sin θ ∙ cos θ) = cos θ / (1/2 sin 2θ)
Applying the reciprocal identity, sin 2θ = 2 sin θ cos θ:
cos θ / (1/2 sin 2θ) = cos θ / (1/2 ⋅ 2 sin θ cos θ)
= cos θ / (sin θ cos θ)
= 1 / sin θ
And finally, using the definition of csc θ from equation (3):
1 / sin θ = csc θ
Hence, we have verified that cot θ ∙ sec θ is equal to csc θ.
To verify the identity cot θ ∙ sec θ = csc θ, we'll manipulate the left-hand side (LHS) of the equation and simplify it to match the right-hand side (RHS):
Starting with LHS:
cot θ ∙ sec θ
We know that cot θ = 1/tan θ and sec θ = 1/cos θ. Substituting these values, we get:
(1/tan θ) ∙ (1/cos θ)
To simplify further, we'll combine the two fractions. Since division is equivalent to multiplying by the reciprocal, we can rewrite 1/tan θ as cot θ and 1/cos θ as sec θ:
cot θ ∙ sec θ = (cot θ/cos θ)
Now, we can use the reciprocal identity for cot θ and rewrite it as cos θ/sin θ:
(cot θ/cos θ) = (cos θ/sin θ ∙ cos θ)
Next, we can simplify further by canceling out the common factor cos θ:
(cos θ/sin θ ∙ cos θ) = 1/sin θ
And since 1/sin θ is equal to csc θ, we can conclude that:
cot θ ∙ sec θ = csc θ
Therefore, the identity cot θ ∙ sec θ = csc θ is verified.