Verify the identity:
(tanx+1)/(secx+cscx) = sinx
multiply through by sec+csc to get
tan+1 = sin(sec+csc)
= sinsec + sincsc
= tan + 1
LS= (sinx/cosx + 1)/(1/cosx + 1/sinx)
= [(sinx + cosx)/cosx] / [ (sinx + cosx)/(sinxcosx)]
= (sinx+ cosx)/cosx * (sinxcosx)/(sinx+cosx)
= sinxcosx/cosx
= sinx
= RS
To verify the identity, we need to simplify the left side of the equation and show that it is equal to the right side.
Starting with the left side of the equation:
(tanx+1)/(secx+cscx)
To simplify this expression, we need to simplify the individual terms in the numerator and denominator.
1. Simplifying the numerator:
The numerator contains tan(x) + 1. We can rewrite tan(x) as sin(x)/cos(x):
sin(x)/cos(x) + 1
To combine these terms, we need a common denominator. The common denominator is cos(x):
(sin(x) + cos(x))/cos(x)
2. Simplifying the denominator:
The denominator contains sec(x) + csc(x). We can rewrite these trigonometric functions as their reciprocal functions:
sec(x) = 1/cos(x)
csc(x) = 1/sin(x)
Replacing the terms in the denominator:
(1/cos(x) + 1/sin(x))
To combine these terms, we need a common denominator. The common denominator is sin(x) * cos(x):
[(sin(x) + cos(x))/(sin(x) * cos(x))]
Now we can rewrite the left side of the equation:
[(sin(x) + cos(x))/(sin(x) * cos(x))] / [(sin(x) + cos(x))/cos(x)]
To divide by a fraction, we can multiply by its reciprocal. In this case, we can multiply by the reciprocal of [(sin(x) + cos(x))/cos(x)]:
[(sin(x) + cos(x))/(sin(x) * cos(x))] * [cos(x)/(sin(x) + cos(x))]
Now, we can cancel out the terms in the numerator and denominator that are the same:
[(sin(x) + cos(x))/(sin(x) * cos(x))] * [cos(x)/(sin(x) + cos(x))]
= (1/(sin(x) * cos(x))) * [cos(x)/ 1]
= cos(x)/(sin(x) * cos(x))
Finally, we can cancel out the cos(x) terms in the numerator and denominator:
cos(x)/(sin(x) * cos(x))
= 1/sin(x)
= csc(x)
Therefore, we have shown that the left side of the equation simplifies to the right side:
(tanx+1)/(secx+cscx) = csc(x)
Hence, the identity is verified.