prove:
sec^2(x)=1+tan^2(x)
RS = 1 + sin^2 x/cos^2 x
= (cos^2 x + sin^2 x)/cos^2 x , using common denominator
= 1/cos^2 x
= sec^2 x
= LS
thanks Reiny
wecome,
please stick to one name, switching names makes it hard to go back in replies
To prove the given trigonometric identity, sec^2(x) = 1 + tan^2(x), we can start from the fundamental trigonometric identity:
sec^2(x) = 1 + tan^2(x)
Step 1:
Recall the definition of secant and tangent functions:
sec(x) = 1/cos(x)
tan(x) = sin(x)/cos(x)
Step 2:
Substitute the definitions of sec(x) and tan(x) into the given identity:
(1/cos(x))^2 = 1 + (sin(x)/cos(x))^2
Step 3:
Square both sides of the equation to eliminate the square root:
(1/cos(x))^2 = 1 + sin^2(x)/cos^2(x)
Step 4:
Multiply both sides of the equation by cos^2(x) to remove the denominator:
1 = cos^2(x) + sin^2(x)
Step 5:
Apply the Pythagorean identity:
1 = 1
Since the left-hand side (LHS) and the right-hand side (RHS) match, the identity is proven true.