Prove the following - sec - - sec - - tan - - tan - sec - - sec - - tan - - tan - - tan -4 - -2tan -4 - -2

L.H.S = (cos θ + cosϕ ) + sin (θ ϕ)(cos θ + cosϕ ) sin (θ ϕ) = cos θ + ϕ2 + sin θ ϕ2cos θ + ϕ2 sin θ ...

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Byju's Answer

Standard XII

Mathematics

Euler's Representation

Prove the fol...

Question

Prove the following :
$\frac{s e c θ + s e c ϕ + (t a n θ - t a n ϕ)}{s e c θ + s e c ϕ - (t a n θ - t a n ϕ)} = \frac{t a n (\frac{π}{4} + \frac{θ}{2})}{t a n (\frac{π}{4} + \frac{ϕ}{2})}$

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Solution

$L . H . S = \frac{(c o s θ + c o s ϕ) + s i n (θ - ϕ)}{(c o s θ + c o s ϕ) - s i n (θ - ϕ)}$
$= \frac{c o s \frac{θ + ϕ}{2} + s i n \frac{θ - ϕ}{2}}{c o s \frac{θ + ϕ}{2} - s i n \frac{θ - ϕ}{2}}$ ....(1)
we have cancelled the common factor 2 cos $\frac{θ - ϕ}{2}$
Now replace cos $\frac{θ + ϕ}{2}$ by sin $[\frac{π}{2} - (\frac{θ - π}{2})]$ and apply sin C $\pm$ sin D etc

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