Verify the identity .
cot ( theta - pi / 2 )= - tan theta
Nevermind for this question
To verify the given identity cot (theta - pi / 2) = -tan theta, we need to simplify both sides of the equation and show that they are equal.
Starting with the left side of the equation, we have cot (theta - pi / 2). The cotangent function is the reciprocal of the tangent function, so we can rewrite this as 1/tan(theta - pi / 2).
Now, using the trigonometric identity tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a) * tan(b)), we can rewrite the denominator of our expression as 1 + tan(theta) * tan(-pi / 2).
The tangent of -pi / 2 is undefined, so tan(-pi / 2) = undefined. Therefore, our expression becomes 1/tan(theta - pi / 2) = 1 / (tan(theta) + undefined).
Since any number divided by undefined is also undefined, the right side of the equation becomes undefined. Hence, we can conclude that cot (theta - pi / 2) is not equal to -tan(theta).
Thus, the given identity cot (theta - pi / 2) = -tan(theta) is not true.
To verify the given identity cot(theta - pi/2) = -tan(theta), we need to simplify both sides of the equation and show that they are equal.
Let's start with the left-hand side (LHS):
cot(theta - pi/2)
In trigonometry, cot(theta) is the reciprocal of tan(theta), so we can rewrite it as:
1 / tan(theta - pi/2)
Using the trigonometric identity tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B)), we can rewrite the expression as:
1 / [(tan(theta) - tan(pi/2)) / (1 + tan(theta)tan(pi/2))]
Since tan(pi/2) is undefined (division by zero), the expression becomes:
1 / [(tan(theta) - undefined) / (1 + tan(theta) * undefined)]
Since multiplying or dividing by undefined gives undefined, the above expression is also undefined.
Now let's simplify the right-hand side (RHS):
-tan(theta)
This is the negative of tan(theta), and it is already simplified.
Comparing the LHS and RHS, we can see that they are NOT equal. Therefore, the given identity cot(theta - pi/2) = -tan(theta) is NOT true.
In this case, the identity provided is incorrect, as the LHS is undefined while the RHS is defined.