Use the cofunction identities to evaluate the expression below without the aid of a calculator.

Question

Quizplus · Accepted Answer

We can use the cofunction identity for sine and cosine, which states that: 
sin(90° - x) = cos(x) and cos(90° -x) = sin(x)
Therefore, 
sin(75°) = cos(90° - 75°) = cos(15°)
Now, we can use the cofunction identity for tangent and cotangent, which states that: 
tan(90° - x) = cot(x) and cot(90° - x) = tan(x)
Therefore, 
tan(75°) = cot(90° - 75°) = cot(15°)
But we also know that cot(x) = 1/tan(x), so: 
cot(15°) = 1/tan(15°)
Now, we can use the cofunction identity for secant and cosecant, which states that: 
sec(90° - x) = csc(x) and csc(90° - x) = sec(x)
Therefore, 
sec(75°) = csc(90° - 75°) = csc(15°)
But we also know that csc(x) = 1/sin(x), so: 
csc(15°) = 1/sin(15°)
Putting it all together: 
sin(75°) + 2tan(75°) - 3sec(75°) 
= cos(15°) + 2cot(15°) - 3csc(15°) 
= cos(15°) + 2(1/tan(15°)) - 3(1/sin(15°))
= cos(15°) + 2cos(15°)/sin(15°) - 3/ sin(15°) 
= (sin(15°)cos(15°) + 2cos(15°) - 3)/sin(15°) 
= [(1/2)sin(30°) + 2(√6 + √2)/4 - 3]/sin(15°) 
= [(1/2)(1/2) + (√6 + √2)/2 - 3]/(1/2)sin(30°) 
= [(1/4) + (√6 + √2)/2 - 3]/(1/2)(1/2) 
= (1 + 2√6 + 2√2 - 12)/2 
= -5 + √6 + √2

Therefore, the answer is B) 2.

Use the Cofunction Identities to Evaluate the Expression Below Without