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PSYCHO15rus [73]
3 years ago
10

Prove the following identity :Sin α . Cos α . Tan α =  (1 – Cos α)  (1 + Cos α)

Mathematics
2 answers:
lyudmila [28]3 years ago
8 0
<span><u><em>Question</em></u>
Sin α . Cos α . Tan α =  (1 – Cos α)  (1 + Cos α)

<u><em>Answer</em></u>
<u><em>Left side</em> </u> = Sin β . Tan β + Cos β 
               = Sin </span>β . Sin β / Cos β + Cos β
               = Sin² β / Cos β + Cos² β / Cos β
               = 1 / Cos β = Sec β = <u><em>Right side proven</em></u>
Umnica [9.8K]3 years ago
4 0
Let's work on the left side first. And remember that
the<u> tangent</u> is the same as <u>sin/cos</u>.

sin(a) cos(a) tan(a)

Substitute for the tangent:

[ sin(a) cos(a) ] [ sin(a)/cos(a) ]

Cancel the cos(a) from the top and bottom, and you're left with

[ sin(a) ] . . . . . [ sin(a) ] which is [ <u>sin²(a)</u> ]  That's the <u>left side</u>.

Now, work on the right side:

[ 1 - cos(a) ] [ 1 + cos(a) ]

Multiply that all out, using FOIL:

[ 1 + cos(a) - cos(a) - cos²(a) ]

= [ <u>1 - cos²(a)</u> ] That's the <u>right side</u>.

Do you remember that for any angle, sin²(b) + cos²(b) = 1  ?
Subtract cos²(b) from each side, and you have sin²(b) = 1 - cos²(b) for any angle.

So, on the <u>right side</u>, you could write [ <u>sin²(a)</u> ] .

Now look back about 9 lines, and compare that to the result we got for the <u>left side</u> .

They look quite similar. In fact, they're identical. And so the identity is proven.

Whew !





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