Question

Question

sec² ø + cosec² ø = sec² ø × cosec² ø

Mathematics
Class 10

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Answer 1

Refer to the attachment for the solution.

Hope it helps!

Answer 2

To Proof:-

• Sec²θ + Cosec²θ = Sec²θ × Cosec²θ

Solution:-

Taking LHS,

We know,

$\dag\blue{\boxed{\underline{\pink{\sf{Sec\theta = \dfrac{1}{Cos\theta}}}}}}$

$\dag\green{\boxed{\underline{\red{\sf{Cosec\theta = \dfrac{1}{Sin\theta}}}}}}$

Hence,

$\sf{Sec^2\theta + Cosec^2\theta}$ can also be written as, $\sf{\dfrac{1}{Cos^2\theta} + \dfrac{1}{Sin^2\theta}}$

$= \sf{\dfrac{1}{Cos^2\theta} + \dfrac{1}{Sin^2\theta}}$

Taking LCM,

$= \sf{\dfrac{Sin^2\theta + Cos^2\theta}{Sin^2\theta \times Cos^2\theta}}$

We know,

$\dag\boxed{\underline{\pink{\sf{Sin^2\theta + Cos^2\theta = 1}}}}$

Hence,

$= \sf{\dfrac{1}{Sin^2\theta Cos^2\theta}}$

$= \sf{\dfrac{1}{Sin^2\theta} \times \dfrac{1}{Cos^2\theta}}$

$= \sf{Cosec^2\theta \times Sec^2\theta}$

$\sf{\implies Sec^2\theta \times Cosec^2\theta}$

Taking RHS,

$\sf{Sec^2\theta \times Cosec^2\theta}$

Hence, Proved!!!

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Things To Remember!!

$\boxed{\begin{array}{c|c|c} \sf{Sin\theta} &\longrightarrow & \sf{\dfrac{1}{Cosec\theta}} \\ \sf{Cos\theta} & \longrightarrow & \sf{\dfrac{1}{Sec\theta}} \\ \sf{Tan\theta} & \longrightarrow & \sf{\dfrac{1}{Cot\theta}} \\ \sf{Cosec\theta} & \longrightarrow & \sf{\dfrac{1}{Sin\theta}} \\ \sf{Sec\theta} & \longrightarrow & \sf{\dfrac{1}{Cos\theta}} \\ \sf{Cot\theta} & \longrightarrow & \sf{\dfrac{1}{Tan\theta}} \end{array}}$

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