verify identity:
(tanx-cotx)/sinxcosx=sec^2x csc^2x
LS = (sinx/cosx - cosx/sinx)/(sinxcosx)
= [(sin^2x - cos^2x)/(sinxcosx)]/sinxcosx
= (sin^2x - cos^2x)/(sin^2x cos^2x)
=sin^2x/(sin^2xcos^2x) - cos^2x/(sin^2xcos^2x)
= 1/cos^2x - 1/sin^2x
= sec^2x - csc^2x
≠ RS
you have a typo, your right side should have been
sec^2x - csc^2x
what is the function of tan 7pi
To verify the given equation:
Step 1: Start with the left-hand side (LHS) of the equation: (tan(x) - cot(x))/(sin(x)cos(x)).
Step 2: Recall the trigonometric identities:
- tan(x) = sin(x)/cos(x)
- cot(x) = cos(x)/sin(x)
- sec^2(x) = 1/cos^2(x)
- csc^2(x) = 1/sin^2(x)
Step 3: Substitute the values of tan(x) and cot(x) from the identities into the LHS of the equation:
LHS = (sin(x)/cos(x) - cos(x)/sin(x))/(sin(x)cos(x))
Step 4: Find the common denominator for the fraction:
LHS = (sin^2(x) - cos^2(x))/(sin(x)cos(x))
Step 5: Use the trigonometric identity sin^2(x) - cos^2(x) = -cos(2x):
LHS = -cos(2x)/(sin(x)cos(x))
Step 6: Apply the double angle formula for cosine: cos(2x) = 2cos^2(x) - 1:
LHS = -(2cos^2(x) - 1)/(sin(x)cos(x))
Step 7: Flip the sign and rearrange the expression to match the right-hand side (RHS):
LHS = (1 - 2cos^2(x))/(sin(x)cos(x)) = -(-1 + 2cos^2(x))/(sin(x)cos(x)) = -((2cos^2(x) - 1))/(sin(x)cos(x)) = -sec^2(x)csc^2(x)
Step 8: The LHS matches the RHS: -sec^2(x)csc^2(x). Therefore, the given equation is verified.
To verify the given identity:
(tanx-cotx)/sinxcosx = sec^2x csc^2x
We can simplify the left side of the equation using basic trigonometric identities, such as the reciprocal identities and the Pythagorean identity.
Starting with the left side:
(tanx - cotx)/(sinxcosx)
Using the reciprocal identities:
(tanx - 1/tanx)/(sinxcosx)
To combine the fractions, we need a common denominator. Multiplying the first fraction by tanx/tanx:
((tan^2x - 1)/tanx)/(sinxcosx)
Using the Pythagorean identity (tan^2x = sec^2x - 1):
((sec^2x - 1)/tanx)/(sinxcosx)
Next, we can simplify the right side of the equation:
sec^2x csc^2x
Using the reciprocal identities:
(1/cos^2x)(1/sin^2x)
Combining the fractions:
1/(cos^2x sin^2x)
Now, we can simplify the left side further:
((sec^2x - 1)/tanx)/(sinxcosx)
Multiplying the numerator and denominator by sin^2x:
((sec^2x - 1)sin^2x)/(tanx sinxcosx)
Using the Pythagorean identity again (1 - sin^2x = cos^2x):
((sec^2x - cos^2x)sin^2x)/(tanx sinxcosx)
Using the reciprocal identities and canceling out common factors:
(sec^2x - cos^2x)/(tanx cosx)
We are left with:
(sec^2x - cos^2x)/(tanx cosx)
Now, we can use the Pythagorean identity (sec^2x = tan^2x + 1) to simplify the numerator:
(tan^2x + 1 - cos^2x)/(tanx cosx)
Combining like terms:
(tan^2x - cos^2x + 1)/(tanx cosx)
Using the difference of squares (tan^2x - cos^2x = sin^2x):
(sin^2x + 1)/(tanx cosx)
Using the Pythagorean identity (sin^2x + cos^2x = 1):
(1 + 1)/(tanx cosx)
Simplifying:
2/(tanx cosx)
Using the reciprocal identities (cotx = 1/tanx):
2/(cotx cosx)
Multiplying the numerator and denominator by sinx:
(2 sinx)/(cotx cosx sinx)
Using the reciprocal identities (cotx = cosx/sinx):
(2 sinx)/(cosx cosx sinx)
Canceling out common factors:
2/(cosx)
This matches the right side of the equation, so we have successfully verified the identity:
(tanx-cotx)/sinxcosx = sec^2x csc^2x.