Prove that sin A1-tan A-cos A1- A- A-cosec A

LHS = sinA(1 + tan A) + cosA(1 + cot A) = sin A (1 + sinA/cosA) + cos A (1 + cosA/sinA) = sin A (cosA+sinA/cosA) + cos A (sinA+cosA/sinA) = sin^2A(cosA + s ...

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Standard X

Mathematics

Trigonometric Ratios

Prove that si...

Question

Prove that sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A

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Solution

LHS = sinA(1 + tan A) + cosA(1 + cot A)

= sin A (1 + $\frac{s i n A}{c o s A}$ ) + cos A (1 + $\frac{c o s A}{s i n A}$ )

= sin A ( $\frac{c o s A + s i n A}{c o s A}$ ) + cos A ( $\frac{s i n A + c o s A}{s i n A}$ )

= $\frac{s i n^{2} A (c o s A + s i n A) + c o s^{2} A (s i n A + c o s A)}{c o s A s i n A}$

= $\frac{(c o s A)}{c o s A s i n A}$ + $\frac{s i n A}{c o s A s i n A}$

= $\frac{1}{s i n A}$ + $\frac{1}{c o s A}$

= cosecA + secA(RHS)

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Prove that sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A

LHS = sinA(1 + tan A) + cosA(1 + cot A) = sin A (1 + sinAcosA) + cos A (1 + cosAsinA) = sin A (cosA+sinAcosA) + cos A (sinA+cosAsinA) = sin2A(cosA+sinA)+cos2A(sinA+cosA)cosAsinA = (cosA)cosAsinA + sinAcosAsinA = 1sinA + 1cosA = cosecA + secA(RHS)