c1 cos2 x +c2 sin2 x +c3 sec2 x +c4 tan2 x = 0when c1 = c2 = 1, c3 = -1, c4 = 1.Example 5CH3_21

The given equation is:

c1 cos²x + c2 sin²x + c3 sec²x + c4 tan²x = 0

Substitute the given values c1 = c2 = 1, c3 = -1, c4 = 1 into the equation:

cos²x + sin²x - sec²x + tan²x = 0

We know that cos²x + sin²x = 1 (identity in trigonometry), and sec²x = 1 + tan²x (another trigonometric identity). Substituting these identities into the equation, we get:

1 - (1 + tan²x) + tan²x = 0

Simplify the equation:

1 - 1 - tan²x + tan²x = 0

0 = 0

This is a true statement, so the original equation holds for all values of x when c1 = c2 = 1, c3 = -1, c4 = 1.