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If sec = 5/4 , show that ( { sin -2cos }{ tan -cos } ) =

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If sec $θ = 5/4$ , show that $(\frac{s in θ - 2 cos θ}{t an θ - cos θ}) = \frac{12}{7}$

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If sec

θ = 5/4

, show that

(\frac{s in θ - 2 cos θ}{t an θ - cos θ}) = \frac{12}{7}

12 mins ago

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12 mins ago

Text solutionVerified

Given: sec

θ = \frac{5}{4}

sec θ = \frac{5 k}{( 4 k )} = \frac{5}{4}

and

cos θ = \frac{4}{5}

Where k is any positive.

By pythagoras theorem:

A C^{2} = A B^{2} + B C^{2}

B C^{2} = A C^{2} - A B^{2}

= 25 k^{2} + 16 k^{2}

BC = 3 k

Find

s in θ, t an θ

and

co tθ

using their definitions:

s in θ = \frac{p er p e n d i c u l a r}{h y p o t e n u se} = \frac{3}{5}

t an θ = \frac{p er p e n d i c u l a r}{ba se} = \frac{3}{4}

co tθ = \frac{1}{t an θ} = \frac{4}{3}

Now,

LHS

= (\frac{s in θ - 2 cos θ}{t an θ - co tθ})

= \frac{\frac{3}{5} - 2 \times \frac{4}{5}}{\frac{3}{4} - \frac{4}{3}}

L H S = \frac{\frac{3 - 8}{5}}{\frac{9 - 16}{12}}

L H S = \frac{12}{7}

L H S = R H S

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Question Text	If sec $θ = 5/4$ , show that $(\frac{s in θ - 2 cos θ}{t an θ - cos θ}) = \frac{12}{7}$
Answer Type	Text solution:1
Upvotes	150