Question

prove that (2+2 sec theta)(1-sec theta)/(2+2cosec theta)(1-cosec theta)=tan 4 theta

Answer 1

Step-by-step explanation:

(2+2sec∅)(1-sec∅)/(2+2cosec∅)(1-cosec∅)

= 2(1+sec∅)(1-sec∅)/2(1+cosec∅)(1-cosec∅)

=2(1-sec²∅)/2(1-cosec²∅)

=-tan²∅/-cot²∅

=tan²∅.tan²∅

=tan⁴∅

Hence Proved.

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Answer 2

Answer:

It is proved that $\frac{(2+2sec\theta)(1-sec\theta)}{(2+2cosec\theta)(1-cosec\theta)}=tan^{4}\theta$

Step-by-step explanation:

The given trigonometric expression is

$\frac{(2+2sec\theta)(1-sec\theta)}{(2+2cosec\theta)(1-cosec\theta)}$

We are required to prove that the value of this expression is equal to $tan^{4}\theta$

We shall do it in the following way as shown below.

$\frac{(2+2sec\theta)(1-sec\theta)}{(2+2cosec\theta)(1-cosec\theta)}\\=\frac{2(1+sec\theta)(1-sec\theta)}{2(1+cosec\theta)(1-cosec\theta)}\\=\frac{1-sec^{2}\theta}{1-cosec^{2}\theta}$

To solve it further we use the trigonometric identities as shown below

$1+tan^{2}\theta=sec^{2}\theta= > 1-sec^{2}\theta=-tan^{2}\theta\\1+cot^{2}\theta=cosec^{2}\theta= > 1-cosec^{2}\theta=-cot^{2}\theta$

Substituting these values in above expression,we get

$=\frac{-tan^{2}\theta}{-cot^{2}\theta} =tan^{4}\theta$

Hence it is proved that,

$\frac{(2+2sec\theta)(1-sec\theta)}{(2+2cosec\theta)(1-cosec\theta)}=tan^{4}\theta$

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