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SOLUTION: Vertify the identity sin(x).sec(x)/sec(x)-1=(sec(x)+1).cot(x)
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-> SOLUTION: Vertify the identity sin(x).sec(x)/sec(x)-1=(sec(x)+1).cot(x)
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Question 401039
:
Vertify the identity
sin(x).sec(x)/sec(x)-1=(sec(x)+1).cot(x)
Answer by
jsmallt9(3757)
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When working with trig equations and you can't figure out the solution, it often helps to convert sec, csc, tan and/or cot to the equivalent expression in terms of sin and/or cos. So this is where we will start:
On the left side we will multiply the numerator and denominator of the "big" fraction by cos(x) to eliminate the "little" fractions. On the right side we just use the Distributive property to multiply:
which simplifies as follows:
Next we can multiply the numerator and denominator of the left side by (1+cos(x)). (You'll see why shortly.)
which simplifies to:
The denominator is equal to sin
^{2}
(x). (This is why we multiplied by 1+cos(x) - it turns the denominator into an expression of sin. We can see that the right side is already in terms of sin so this development is a positive one.)
A factor of sin(x) cancels on the left side:
leaving
The left side is now exactly the same as the right side so we have proven the identity.