Question
Question: How do you solve \(\dfrac{1+\cos x}{1+\sec x}\) ?...
How do you solve 1+secx1+cosx ?
Solution
To solve the given trigonometric expression i.e. 1+secx1+cosx, we are going to use the property of secx which is equal to the reciprocal of cosx and mathematically we can write secx=cosx1 in the above expression and then simplify it.
Complete step by step answer:
The trigonometric expression given in the above problem is as follows:
1+secx1+cosx
We know the relation between secx&cosx is equal to:
secx=cosx1
Using the above relation in the given trigonometric expression we get,
⇒1+cosx11+cosx
Now, taking L.C.M of cosx in the denominator in the above expression we get,
⇒cosxcosx+11+cosx
Rearranging the above trigonometric expression we get,
⇒1+cosxcosx(1+cosx)
In the above expression, (1+cosx) is common in the numerator and denominator so (1+cosx) will be cancelled out from the numerator and the denominator and we get,
⇒cosx
From the above simplification of the given trigonometric expression we get,
⇒cosx
Note: You might think how we know whether to use secx=cosx1, the answer is in the numerator you can see there is a term cosx so if we can use this secx&cosx relation then there is a possibility in which cosx term might get cancelled and which you can see is the case in the above solution when we use this relation (1+cosx) in the numerator and the denominator will be cancelled out.
The similar problem which can be possible is as follows:
1+cosecx1+sinx
So, here we can use the relation between cosecx=sinx1 and then simplify and we get,
⇒1+sinx11+sinx
Taking sinx as L.C.M in the denominator of the above expression we get,
⇒sinxsinx+11+sinx=1+sinxsinx(1+sinx)
In the above expression, (1+sinx) is common in numerator and the denominator so we can cancel this expression and we get,
⇒sinx