Simplify secX+cscX/1+tanX
- Thread starter adpcane15
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Does the above mean either of the following?adpcane15 said:
. . . . .sec(x) + [ csc(x) / (1 + tan(x)) ]
. . . . .[ sec(x) + csc(x) ] / [ 1 + tan(x) ]
Or something else?
What have you tried so far?
Please be specific. Thank you.
Eliz.
Hello, adpcane15!
Convert to sines and cosines: \(\displaystyle \L\:\frac{\frac{1}{\cos x}\,+\,\frac{1}{\sin x}}{1\,+\,\frac{\sin x}{\cos x}}\)
Multply top and bottom by \(\displaystyle \sin x\cdot\cos x\)
. . \(\displaystyle \L\frac{\sin x\cdot\cos x}{\sin x\cdot\cos x}\,\cdot\,\frac{\frac{1}{\cos x}\,+\,\frac{1}{\sin x}}{1\,+\,\frac{\sin x}{\cos x}} \;=\;\frac{\sin x\,+\,\cos x}{\sin x\cdot\cos x\,+\,\sin^2x}\)
Factor and reduce: \(\displaystyle \L\:\frac{\sout{\sin x\,+\,\cos x}}{\sin x(\sout{\cos x\,+\,\sin x})} \;=\;\frac{1}{\sin x} \;=\;\csc x\)
Convert to sines and cosines: \(\displaystyle \L\:\frac{\frac{1}{\cos x}\,+\,\frac{1}{\sin x}}{1\,+\,\frac{\sin x}{\cos x}}\)
Multply top and bottom by \(\displaystyle \sin x\cdot\cos x\)
. . \(\displaystyle \L\frac{\sin x\cdot\cos x}{\sin x\cdot\cos x}\,\cdot\,\frac{\frac{1}{\cos x}\,+\,\frac{1}{\sin x}}{1\,+\,\frac{\sin x}{\cos x}} \;=\;\frac{\sin x\,+\,\cos x}{\sin x\cdot\cos x\,+\,\sin^2x}\)
Factor and reduce: \(\displaystyle \L\:\frac{\sout{\sin x\,+\,\cos x}}{\sin x(\sout{\cos x\,+\,\sin x})} \;=\;\frac{1}{\sin x} \;=\;\csc x\)