Evaluate tan B tan C if triangle ABC is not a right trianlge and cos A=cosB cosC.
To evaluate tan B tan C, we can use the given information that triangle ABC is not a right triangle and cos A = cos B cos C.
First, let's visualize a non-right triangle ABC and label the angles A, B, and C.
Since cos A = cos B cos C, we can conclude that angle A is equal to the product of angles B and C.
Now, recall the trigonometric identity:
tan θ = sin θ / cos θ
We can rewrite tan B tan C as:
tan B tan C = (sin B / cos B) * (sin C / cos C)
Using the relationship between sine and cosine:
sin² θ + cos² θ = 1
We can express the terms sin B / cos B and sin C / cos C in terms of sine alone:
sin B / cos B = sqrt(1 - cos² B) / cos B
sin C / cos C = sqrt(1 - cos² C) / cos C
Substituting these expressions back into tan B tan C:
tan B tan C = (sqrt(1 - cos² B) / cos B) * (sqrt(1 - cos² C) / cos C)
Now, using the given information that angle A is equal to the product of angles B and C, we can substitute cos A = cos B cos C:
cos A = cos B cos C
cos A = cos² B
cos B = sqrt(cos A)
Substituting this back into our expression:
tan B tan C = (sqrt(1 - sqrt(cos A)²) / sqrt(cos A)) * (sqrt(1 - cos² C) / cos C)
Simplifying this expression results in the evaluation of tan B tan C specific to the given conditions.