Question

If sec θ +tan θ =p, show that : Sinθ = (p² -1)/(p²+1)

Answer 1

Solution:

LHS = p²-1/ p²+1


= (secθ +tanθ)²-1 / (secθ +tanθ)²+1


= sec²θ +tan²θ +2secθtanθ-1/sec²θ +tan²θ +2secθtanθ+1



[( a+b)²= a²+b²+2ab]


= (sec²θ -1)+tan²θ +2secθ tanθ /sec²θ +2secθtanθ +(1+tan²θ)



= tan²θ+tan²θ +2secθ tanθ / sec²θ+2secθtanθ +sec²θ


[tan²θ= sec²θ-1] [sec²θ=1+tan²θ]



= 2tan²θ +2secθ tanθ /2 sec²θ+2secθtanθ +sec²θ


= 2tanθ(tanθ+ secθ) / 2secθ(secθ+tanθ)


= tanθ/ cosθ = sinθ


LHS= RHS

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Hope this will help you....