prove that sec(3 pie/2 -x) sec(x-5 pie/2) + tan (5 pie/2+x)tan(x-3 pie/2)=-1
To prove the given trigonometric identity, we need to simplify the left-hand side of the equation until it is equal to -1.
Starting with the left-hand side of the equation:
sec(3π/2 - x) sec(x - 5π/2) + tan(5π/2 + x) tan(x - 3π/2)
First, let's rewrite the trigonometric functions using their reciprocal identities:
1/cos(3π/2 - x) * 1/cos(x - 5π/2) + sin(5π/2 + x)/cos(5π/2 + x) * sin(x - 3π/2)/cos(x - 3π/2)
Next, we can simplify the expression by expanding the reciprocals:
(1/cos(3π/2 - x)) * (1/cos(x - 5π/2)) + (sin(5π/2 + x) * sin(x - 3π/2)) / (cos(5π/2 + x) * cos(x - 3π/2))
Now, let's simplify each term separately:
Term 1:
1/cos(3π/2 - x) = 1/sin(x)
1/cos(x - 5π/2) = 1/sin(π/2 - x) = 1/cos(x)
Therefore,
(1/cos(3π/2 - x)) * (1/cos(x - 5π/2)) = (1/sin(x)) * (1/cos(x)) = 1/(sin(x) * cos(x)) = 1/(sin(x) * cos(x)) * (cos(x) / cos(x)) = cos(x) / (sin(x) * cos(x)) = 1/sin(x)
Term 2:
sin(5π/2 + x) = sin(π/2 + x) = cos(x)
sin(x - 3π/2) = sin(x + π/2) = cos(x)
Therefore,
(sin(5π/2 + x) * sin(x - 3π/2)) / (cos(5π/2 + x) * cos(x - 3π/2)) = (cos(x) * cos(x)) / (cos(x) * cos(x)) = 1
Combining Term 1 and Term 2 back into the left-hand side of the equation:
1/sin(x) + 1 = (1 + sin(x))/sin(x)
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite (1 + sin(x)) as cos^2(x):
(cos^2(x))/sin(x) = cos^2(x)/sin(x)
Since cos(x)/sin(x) is equal to cot(x), we have:
cos^2(x)/sin(x) = cos^2(x) * cot(x)
Using the identity cot(x) = cos(x)/sin(x), we now have:
cos^2(x) * cot(x) = cos^2(x) * cos(x)/sin(x) = cos^3(x)/sin(x)
Finally, using the identity sin^2(x) + cos^2(x) = 1, we can rewrite sin(x) as √(1 - cos^2(x)):
cos^3(x)/sin(x) = cos^3(x)/(√(1 - cos^2(x)))
Simplifying this expression, we get:
= cos^3(x)/sin(x) = cos^3(x)/(√(1 - cos^2(x))) = cos^3(x)/(√(sin^2(x)))
Therefore, the left-hand side simplifies to:
cos^3(x)/(√(sin^2(x)))
Now, let's simplify the right-hand side of the equation:
The right-hand side is -1, which is already in its simplest form.
Therefore, the left-hand side of the equation simplifies to cos^3(x)/(√(sin^2(x))), while the right-hand side is -1, which implies they are not equal.
Hence, the original given statement is not true.